Beauty and the Beast in Mean Curvature Flow Without Singularities
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Beauty and the Beast in Mean Curvature Flow Without Singularities Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften vorgelegt von Maurer, Wolfgang an der Mathematisch-Naturwissenschaftliche Sektion Fachbereich Mathematik und Statistik Konstanz, 2021 Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-2-17lfpln8mq5cv3
Tag der mündlichen Prüfung: 14.01.2021 1. Referent: Prof. Oliver Schnürer 2. Referent: Prof. Miles Simon
Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Project number 336454636.
Für
Deutsche Zusammenfassung Die Dissertation befasst sich mit einem Thema der geometrischen Analysis. Hauptsächlich beschäftigt sie sich mit dem mittleren Krümmungsfluss. Das ist eine geometrische Evolu- tionsgleichung für Hyperflächen, die für die Normalengeschwindigkeit der sich bewegenden Hyperfläche die mittlere Krümmung vorschreibt (oder je nach Vorzeichenkonvention die ne- gative mittlere Krümmung). Diese Gleichung ist aus einer Reihe von Gründen besonders in- teressant. Beispielsweise ist der mittlere Krümmungsfluss der negative 2 -Gradientenfluss des Oberflächenfunktionals. In diesem Sinne verkleinert der mittlere Krümmungsfluss den Flächeninhalt einer Hyperfläche maximal effizient. Der mittlere Krümmungsfluss hat die Angewohnheit, Singularitäten auszubilden. An solchen Singularitäten bricht die klassische Beschreibung der Hyperfläche mittels Parame- trisierungen zusammen und man muss zu sogenannten schwachen Lösungsbegriffen überge- hen. Hierbei gibt es verschiedene Ansätze, auf die wir an dieser Stelle nicht näher eingehen. Ausgangspunkt für uns ist [SS15]. Dort wird die Existenz eines mittleren Krümmungs- flusses von vollständigen graphischen Hyperflächen gezeigt. Diese Lösungen weisen keine Singularitäten auf endlicher Höhe auf. Der Rand der Projektion einer solchen Lösung – oder des Definitionsgebietes des Graphen, wenn man so will – lässt sich als schwache Lösung des mittleren Krümmungsflusses in einer Dimension niedriger interpretieren. Man hat also die vollständige graphische Hyperfläche, die man klassisch, ohne auftretende Singularitäten fließen lassen kann, und in der Projektion liefert der Rand einen mittleren Krümmungs- fluss, der durch Singularitäten hindurch fließt. Die Heuristik ist dabei die Folgende. Die graphische Lösung, definiert über Ω für Zeiten , ist weit oben (oder unten) asymptotisch zur Zylindermenge Ω × R. Weil die graphische Lösung den mittleren Krümmungsfluss erfüllt, erwarten wir dies auch von Ω × R. Weil der zusätzliche Faktor R nicht zur Dyna- mik beiträgt, erwarten wir also, dass bereits Ω , der Rand der Projektion, den mittleren Krümmungsfluss in einer Dimension niedriger erfüllt. Kapitel 2 greift den mittleren Krüm- mungsfluss ohne Singularitäten, wie Sáez und Schnürer ihre Lösungen in [SS15] getauft haben, noch einmal auf und wiederholt die für diese Arbeit wesentlichen Punkte. Ein Ziel der Arbeit ist es, die obige Heuristik rigoros zu machen. Dabei wird in Kapitel 4 erstmal ein negatives Resultat erbracht. Mithilfe einer dort eingeführten Barriere wird ein Beispiel konstruiert, bei dem sich die anfängliche (graphische) Fläche immer wieder umstülpt und nahe am Zylinder immer mehr Schichten ausbildet. Dieses Verhalten bleibt über die ganze Zeit bestehen. Insbesondere ist die graphische Fläche zu keiner Zeit in einer starken Art und Weise zum Zylinder asymptotisch. Die Heuristik ist also nicht ganz zutreffend. Allerdings beobachtet man, dass das immer engere Umstülpen der graphischen Fläche in diesem Beispiel dessen Krümmung unbeschränkt macht. Starten wir mit einer graphischen Fläche beschränkter Krümmung, können wir eine positive Aussage machen. In Kapitel 5 wird gezeigt, dass unter dieser Voraussetzung die graphische Fläche glatt asymptotisch zur Zylindermenge ist, solange die Zylindermenge glatt bleibt, also nicht singulär wird. In Kapitel 6 wird dieses Resultat über Singularitäten der Zylindermenge ii
Deutsche Zusammenfassung hinweg ausgeweitet. Dabei wird allerdings die zusätzliche Annahme gemacht, dass der graphische mittlere Krümmungsfluss -nichtkollabiert ist. Im selben Kapitel zeigen wir auch die Existenz einer solchen -nichtkollabierten, vollständigen, graphischen Lösung des mittleren Krümmungsflusses, die von einer entsprechenden, gegebenen Hyperfläche startet. Neben den Untersuchungen zur Heuristik des mittleren Krümmungsflusses ohne Sin- gularitäten beinhaltet die Arbeit auch eine Anwendung des mittleren Krümmungsflusses ohne Singularitäten. Die Idee einer schwachen Lösung, die von einer klassischen, graphi- schen Lösung im Hintergrund kommt, wird genutzt, um zu zeigen, dass Hyperflächen, die spiegelsymmetrisch zu einer Hyperebene und beiderseits graphisch über dieser Hyperebe- ne sind, unter diesem schwachen Fluss Singularitäten nur auf der Symmetriehyperebene ausbilden. Als Korollar erhält man die Lösung eines freien Randwertproblems, bei dem der Rand der fließenden Hyperfläche eine Hyperebene senkrecht trifft. Schließlich wird in Kapitel 7 die Existenz von vollständigen graphischen Lösungen des -Flusses gezeigt. Beim -Fluss ist die Normalengeschwindigkeit der sich bewegenden Hyperfläche − , wobei die mittlere Krümmung bezeichnet und > 0 ein Exponent ist. Natürlicherweise betrachten wir in diesem Zusammenhang nur Hyperflächen positiver mittlerer Krümmung ( > 0). Mit dem -Fluss betrachten wir eine voll-nichtlineare Evolutionsgleichung, die eine Homogenität verschieden von Eins aufweist. In den letzten Jahren wurde die Existenz vollständiger graphischer Lösungen für immer mehr Normalen- geschwindigkeiten gezeigt [CD16; CDKL19; Xia16; LL19; Hol14; AS15]. Allerdings werden allgemein nur konvexe Lösungen betrachtet. In der vorliegenden Arbeit wird hingegen nur positive mittlere Krümmung verlangt. Dadurch muss mehr Aufwand für die Approximation von Lösungen betrieben werden. Im Anhang werden weitgehend unabhängige Resultate, die in der Arbeit Verwendung finden, erarbeitet oder aus der bestehenden Literatur erschlossen. Oberstes Ziel der Promotion soll der Nachweis der Befähigung zur selbständigen wissen- schaftlichen Arbeit sein. Vor dem Hintergrund der hohen Internationalität der Mathema- tik durch ihre Allgemeingültigkeit und aufgrund der Tatsache, dass der wissenschaftliche Austausch in diesem Fach fast ausschließlich in englischer Sprache geschieht, erscheint es folgerichtig, die Dissertation in englischer Sprache zu verfassen. Dementsprechend habe ich gehandelt. Gemäß den Regularien der Promotionsordnung füge ich aber diese deutsche Zusammenfassung hinzu. Die Danksagung ist ebenfalls in deutscher Sprache gehalten. iii
Danksagung Vor allen anderen möchte ich meinem Betreuer Prof. Dr. Oliver C. Schnürer für die gute Zusammenarbeit und seine volle Unterstützung danken. Er hat meine mathematische Entwicklung schon seit Beginn meines Studiums begleitet und ich hatte das große Glück, von seinen Kenntnissen profitieren zu dürfen. Besonders haben mich dabei die Präzision und Sorgfalt, die er an den Tag legt, beeindruckt. Einen besonderen Dank will ich Herrn Clemens Hauser aussprechen. Sein unschätzba- res Engagement für die Mathe-AG an meinem Gymnasium ist alles andere als selbstver- ständlich. Durch seine Mathe-AG hat er in mir die Liebe zur Mathematik geweckt, was mich dazu geführt hat, den Weg in die Mathematik einzuschlagen – eine Entscheidung, die ich nie bereut habe. Unverzichtbar auf meinem Weg waren meine Eltern. Ohne ihre Unterstützung wäre wohl einiges schwierig geworden und ich hätte sicher nicht die Freiheit und das Vergnügen gehabt, mich in dem Ausmaß mit Mathematik zu beschäftigen. Vielen Dank! Zu guter Letzt will ich der DFG danken, die durch ihr Schwerpunktprogramm „SPP 2026 – Geometry at Infinity” Mittel zur Verfügung gestellt hat, durch die meine Stelle finanziert werden konnte. iv
Contents Deutsche Zusammenfassung iii Danksagung v Notation viii 1. Introduction 1 2. Mean Curvature Flow Without Singularities 5 3. Vain Mean Curvature Flow 8 3.1. Vanity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2. Vain Mean Curvature Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4. Barrier Over an Annulus 16 4.1. Construction of the Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2. Example with Wild Oscillations . . . . . . . . . . . . . . . . . . . . . . . . 23 4.3. Relation Between Spatial and Temporal Asymptotics . . . . . . . . . . . . 25 5. Curvature Bounds 27 5.1. Impossibility of a Controlled Curvature Bound . . . . . . . . . . . . . . . . 27 5.2. Smooth Asymptotics to the Cylinder . . . . . . . . . . . . . . . . . . . . . 28 6. α-Noncollapsed MCF Without Singularities 32 6.1. Asymptotics to the Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.2. Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 α 7. H -Flow Without Singularities 38 7.1. Local A Priori Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 7.2. Auxiliary Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 α 7.3. H -Flow of Complete Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 52 A. Normal Graphs 54 A.1. Geometry of Normal Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 54 A.2. Normal Graphs and Local Graph Representations . . . . . . . . . . . . . . 55 A.3. Hypersurfaces Close to Each Other . . . . . . . . . . . . . . . . . . . . . . 57 B. Position-Dependent Curvature Flows 61 B.1. Curvature Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 B.2. Tensor Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 v
Contents B.3. Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 C. Set flow, Domain flow, and α-Noncollapsed Mean Curvature Flow 69 Bibliography 72 vi
Notation
Set theoretic notation For the ease of notation, we frequently write expressions like
{ > 0} or { +1 > 0}. These are to be read as “the set of all with ( ) > 0” or “the
set of all points with +1 > 0”, respectively. When bearing in mind this flexibility of
notation, it should be clear from the context how such expressions must be interpreted
when they appear.
Asymptotics For 0 ∈ R ∪ {∞} we write
( ) ≃ ( ) ( → 0 )
if
( ) → 0
−−−→ 1 .
( )
( ) ( ) ( )
Because of ℎ( ) = ( ) ℎ( ) , the relation “≃” is transitive.
Einstein summation convention As usual in the field of differential geometry, we use
Einstein summation convention: We implicitely sum over repeated indices appearing in a
term as an lower and upper index.
– the dimension; a natural number ≥ 1
, , , … – indices ranging over 1, …,
, , , … – indices ranging over 1, …, + 1 (especially may have a different meaning
depending on the context)
– Kronecker delta
– standard basis vector of R ; = (0, …, 1, …0)
Derivatives
, – partial derivative with respect to the th coordinate; partial derivative with respect
to
∇ – the covariant derivative from Riemannian geometry. It should always be clear from
the domain and codomain of the function that is being derived which Levi-Civita-
connections must be used. As usual, we take the “covariant” derivative for maps
between the base manifolds as the tangential, so no actual connection is being used
then.
viiNotation
, – short-hand for ∇ , ∇2
∇∗ – the gradient operator corresponding to the covariant derivative and the given metric
(the covariant derivative with a raised index)
If a time variable is involved, derivatives denoted with ∇ or always refer to the spatial
variable only. If a derivative with respect to both temporal and spatial variables is meant,
we denote it with ∇( , ) .
Hypersurfaces
– an immersion or parametrization of a hypersurface
– normal vector; by default we choose the outwards or downwards pointing normal
– coefficients of the induced metric
– coefficients of the inverse metric
ℎ – coefficients of the second fundamental form with sign convention such that ℎ =
−⟨ , ∇2 ⟩
| | – norm of the second fundamental form; given by | |2 = ℎ ℎ
– mean curvature; given by ℎ
Curvature flows without singularities
– spatial variable/coordinate
– time variable/coordinate
– the function of ( , ) that satisfies the graphical curvature flow
Ω – the domain of definition for ; a relatively open subset of R × [0, ∞)
Ω – the -time-slice of Ω such that Ω = ⋃ Ω × { }; an open subset of R
– the hypersurface graph (⋅, )
R – The extended real line R∪{±∞} with the natural topology (homeomorphic to [−1, 1]).
viii1. Introduction Motivation. The topic of this thesis belongs to the field of geometric analysis. The thesis is mostly concerned with mean curvature flow, which is a curvature driven evolution of a hypersurface. Mean curvature flow is interesting for a number of reasons. Firstly, it is the negative 2 -gradient flow of the area functional. In this sense, mean curvature flow seeks to minimize the area of the hypersurface in the most efficient way. For this reason, mean curvature flow or variants of it are used for modeling physical phenomena where the energy of a surface is proportional to its area. For instance, an often cited example is motion of grain boundaries in materials science [Mul56]. But this gradient flow is interesting from a purely mathematical point of view, too, and can lead to proofs of isoperimetric inequalities, e.g., [Sch08]. There is yet another reason why the mean curvature flow is important for differential geometry. In a sense, it is the simplest geometric evolution equation of parabolic type for hypersurfaces. (I will elaborate on this in a moment.) As such, mean curvature flow is a convenient tool to deform hypersurfaces. Building on this idea, mean curvature flow with surgery has been used to obtain a topological structure theorem for 2-convex hypersurfaces [HS09; BH13; BH18; HK17]. These results mirror more famous ones using the Ricci flow, an intrinsic flow which deforms a metric on a manifold. Hamilton’s program led to a proof of the Poincaré Conjecture and Thurston’s Geometrization Conjecture by Perelman using 3D Ricci flow with surgery [Per03a; Per03b; Per02]. As the Poincaré Conjecture was one of the “Millennium Problems,” its resolution drew attention even beyond the mathematical world (arguably, Perelman’s rejection of the awards did even more). Description of MCF. Back to mean curvature flow (MCF): A hypersurface is said to move by its mean curvature if the normal velocity of the evolving hypersurface is given by minus the mean curvature (or the mean curvature depending on the sign convention). If ∶ × → R +1 is an immersion of a moving hypersurface = im (⋅, ), say, with a time interval. Then moves by its mean curvature, or is said to be a solution of mean curvature flow, if and only if ⟨ , ⟩R +1 = − . (1.1) MCF is described by a second order parabolic partial differential equation (PDE). This becomes most apparent if one parametrizes along the normal direction, ∥ . Then (1.1) takes the form = Δ , where Δ is the Laplace-Beltrami-operator with respect to the induced metric. So MCF resembles the heat equation, which is the prototypical example of a parabolic PDE. However, this argument is hocus-pocus to some extend: Although “ = Δ ” looks nice and is geometrically appealing, this equation is awkward from a PDE point of view: It is a degenerate parabolic system. This stems from the fact that in this equation the mere act of taking a derivative ∇ is dependent on the solution . The degeneracy arises because the Christoffel symbols carry second derivatives of to the 1
1. Introduction equation. Actually, in coordinates we obtain Δ = ( − ) , (1.2) which can easily be seen to be degenerate in tangential directions by multiplying the coefficient “matrix” with . A way out is what is known as “DeTurck’s Trick” or as “fixing a gauge”: One fixes an arbitrary (time-dependent) covariant derivative on and realizes that (1.1) can be written with this derivative: ⟨ , ⟩ = ⟨ , 2 ⟩ . (1.3) In particular, for = 2 (1.4) is a solution of MCF, i.e., solves (1.1). Equation (1.4), in turn, is a solid parabo- lic PDE system. Because the induced metric is dependent on , equation (1.4) is a quasilinear parabolic PDE system for . The fact that (1.4) is quasilinear makes MCF simpler than most other geometric flows, which are usually fully nonlinear. If desired, a solution of = Δ can be obtained from a solution of (1.4) by taking time-dependent reparametrizations which one obtains solving ordinary differential equations. Graphical case and preceeding works. In this thesis, graphical hypersurfaces play a major role. These are hypersurfaces which are given as the graph of a function . More concretely, if Ω ⊂ R × is a spacetime domain and ≡ ( , ) is a function defined on Ω, then a parametrization of the evolving hypersurface = graph (⋅, ) is given by ( , ) = ( , ( , )). Taking to be the standard derivative on Ω ⊂ R ( = ), equation (1.4) reduces to = 2 = ( − 2 ) . (1.5) 1 + | |2 This is a scalar quasilinear parabolic PDE of second order and we refer to it as graphical mean curvature flow (GMCF). Initial boundary value problems for this equation have been studied in [Hui89]. In [EH91], local interior estimates for GMCF have been established. With the help of these, in the same paper, existence of solutions of MCF for entire graphs was shown. Taking us further, a similar result for complete graphs was proven in [SS15]. A graph is complete if it is complete as a submanifold. In particular, the representing function ≡ ( , ) need not be defined on all of R × but only on a subset Ω thereof. Moreover, they were able to interpret ( Ω ) as a weak solution to mean curvature flow of one dimension lower. For the weak solution singularities can occur, whereas is completely smooth. They therefore named their scheme “mean curvature flow without singularities.” The heuristic idea is that the complete hypersurface = graph (⋅, ) solving MCF is asymptotic to Ω × R. So one expects Ω to move by MCF, as well, because the flat factor R does not contribute to the dynamics. As the concept of mean curvature flow without singularities is central to this thesis, we will give a short introduction in chapter 2. 2
1. Introduction
Results. This thesis builds on the results of [SS15], i.e., on the mean curvature flow
without singularities. One of the ideas of mean curvature flow without singularities is
to provide a notion of weak solution to MCF which is backed by a smooth graphical
solution. We give an application of this idea in chapter 3. We consider hypersurfaces
with a property we have named “vanity”. This property implies that the hypersurface
is mirror-symmetric to a hyperplane. Moreover, any point can “see” its mirror-image,
hence the name. A vain hypersurface can be flown by the weak mean curvature flow
which is realized as Ω of a mean curvature flow without singularities. The vanity can be
preserved along that flow. The main result of chapter 3 states that all singularities occur
on the hyperplane of symmetry. The proof is completely in the spirit of mean curvature
flow without singularities and works with the smooth hypersurface where one can work
classically. Results on are then brought back to the boundary of its projection Ω , the
weak flow we are interested in. A by-product is a solution of a free boundary value problem
for GMCF where one considers a graphical solution over a hyperplane which meets this
hyperplane perpendicularly. The solution is smooth in the interior but may be singular
towards the boundary.
Having seen an application of mean curvature flow without singularities in chapter 3,
we proceed with an investigation of mean curvature flow without singularities itself. We
inspect the heuristics mentioned above that is asymptotic to Ω ×R. We have achieved
contrary results. In chapter 4 a barrier is provided which allows to construct a rotationally
symmetric solution over a ball which sheets infinitely towards Ω × R and retains this
behavior for all times of existence. As a consequence, ∩{ +1 > } is not even graphical
over Ω × ( , ∞) for any ∈ R, let alone smoothly asymptotic. The mentioned barrier
also allows to prove a certain relationship between the asymptotics of ( , 0) for → Ω0
and (0, ) for → , where = sup{ ∶ Ω ≠ ∅}.
The example in chapter 4 shows that the heuristics that is asymptotic to Ω × R
(in a strong sense) may fail. Nevertheless, in chapter 5 we give an affirmative answer
if the curvature is bounded. Theorem 5.2 states that under the hypothesis that 0 has
bounded curvature, is smoothly asymptotic to Ω × R for positive times as long as the
latter does not become singular. In chapter 6 we extend this result beyond singularities
of Ω × R when we work with the class of -noncollapsed mean convex hypersurfaces.
Appendix C contains a short introduction to -noncollapsed mean curvature flow following
[HK16]. The main take-away is that in this class the mean curvature bounds all of the
terms |∇ | ( = 0, 1, …). This fact allows us to prove curvature bounds on the spacetime
track of and these, in turn, imply the smooth convergence of to Ω × R for all
positive times when Ω × R is smooth.
After our considerations of mean curvature flow without singularities we enter the fully
nonlinear realm in chapter 7. There, we consider the -flow and in place of (1.1) we have
⟨ , ⟩ = − . (1.6)
Herein, > 0 is an arbitrary positive real number. The case = 1 amounts to MCF. For
≠ 1 the resulting equations are fully nonlinear. Furthermore, they are not of homogeneity
one. Naturally, we will only consider mean convex hypersurfaces such that > 0 and
is well-defined. The main theorem of the chapter is existence of complete graphical
solutions of -flow. Entire graphs have been studied in [Fra11].
31. Introduction Open Problems. Starting with the last chapter, it would be nice to allow normal speeds more general than . A class that contained powers of all elementary symmetric functions would be satisfying. Recently, there has been made progress in this direction [CD16; CDKL19; Xia16; LL19; Hol14; AS15]. The class of curvature functions for which existence of complete graphical solutions has been shown has widened a lot. Unfortunately, all of these works only consider convex hypersurfaces instead of the more natural assumption of admissible hypersurfaces. (In [Xia16], the estimates on homogeneity one curvature functions are done for non-convex hypersurfaces, too, but an appropriate approximation scheme is lacking.) With respect to the question of the asymptotics of to Ω ×R, it would be interesting to know if Theorem 5.2 generally holds beyond singularities of Ω even without the additional assumption of -noncollapsedness. The problem is that the curvature of becomes unbounded at a singular time and we know from the example in chapter 4 that Theorem 5.2 may fail under this circumstance. Finally, it would be nice to see more applications of the mean curvature flow without singularities (or another curvature flow without singularities) that lean on the idea of a weak flow with a smooth graphical flow in the background. The elliptic regularization scheme of Ilmanen [Ilm94] shares some similarities in that regard. Elliptic regularization has proven to be very useful and I hope to see mean curvature flow without singularities, too, demonstrate its utility in one or the other context. 4
2. Recapitulation of Mean Curvature
Flow Without Singularities
Since this thesis builds on the ideas of [SS15], it is worthwhile to summarize the main
points of “Mean curvature flow without singularites”. This chapter doesn’t contain new
results. (We follow [SS15], or [Mau16] when we diverge from [SS15].) Instead, it helps
setting the notation and allows us to shorten our exposition lateron when similar steps as
here are needed to be taken. We would also like to point to the notation chapter where a
paragraph is devoted to curvature flows without singularities.
Mean curvature flow without singularities is mainly about the mean curvature flow of
complete graphical hypersurfaces.
(i) Initial Data: Let Ω0 ⊂ R be open and let 0 ∶ Ω0 → R be a locally Lipschitz-
continous function. We assume that there is a continuous extension 0 ∶ R → R of
0 such that { 0 ∈ R} = Ω0 and 0 |Ω0 = hold.
(ii) Solution Data: A mean curvature flow without singularities (aka singularity re-
solving solution to mean curvature flow) is a pair ( , Ω) of an relatively open subset
Ω ⊂ R × [0, ∞) and a continuous function ∶ Ω → R. The zero time slice of Ω is
supposed to be Ω0 , in line with a consistent notation. Moreover, we suppose that
(⋅, 0) = 0 holds. For this reason, we call ( 0 , Ω0 ) the initial data for ( , Ω).
Maximality condition: We suppose that there exists a continuous function ∶ R ×
[0, ∞) → R such that { ∈ R} = Ω and |Ω = hold.
Equation: The function is supposed to be smooth and to satisfy the equation of
graphical mean curvature flow on Ω ∖ (Ω0 × {0}), i.e.,
= ( − ) . (2.1)
1 + | |2
(iii) Hypersurfaces: We denote by ≔ graph (⋅, ) the graphical hypersurfaces that
move by their mean curvature (locally in a classical sense).
(iv) Shadow flow: The family (Ω ) ≥0 is called the shadow flow.
Theorem 2.1. For any such initial data ( 0 , Ω0 ), there exists a corresponding mean
curvature flow without singularities ( , Ω). The shadow flow is a weak solution of mean
curvature flow in dimension − 1 in a sense explained below (Remark 2.2 (iii)).
52. Mean Curvature Flow Without Singularities
Remark 2.2.
(i) The maximality condition implies | ( , )| → ∞ for ( , ) → Ω. ( Ω denotes the
relative boundary of Ω in R × [0, ∞).) In particular, the hypersurfaces are
complete. Moreover, the maximality condition implies that the solution is maximal
in ; stopping the flow at an arbitrary time may prevent the maximality condition to
hold.
The maximality condition is defined slightly differently in [SS15]. Only positive
and proper functions are considered there. We follow [Mau16] here, where these
assumption are dropped and the maximality condition is adapted accordingly.
(ii) Although is smooth, the formalism allows for changes of the topology of .
Singularities of Ω may be interpreted as singularities of at infinity.
(iii) In [SS15], the shadow flow is advertised as a weak solution. They underpin this by
showing that (for their solution) (Ω ) ∈[0,∞) coincides with the level-set flow starting
from Ω0 H -almost everywhere if the level-set flow is not fattening.
In [Mau16], for an arbitrary solution ( , Ω) the shadow flow is interpreted as a weak
solution in the sense of a domain flow (cf. Definition C.4).
Discussion of the proof of Theorem 2.1. One constructs an approximating sequence of func-
tions ∶ R × [0, ∞) → R. It needs to satisfy the following form of local equicontinuity:
For any ∈ R, any ∈ R, and any > 0 there is = ( , , ) > 0 and an index
= ( , , ) ∈ N such that for any ≥ and any ( , ), ( , ) ∈ R × [0, ∞) with
| | < , | ( , )| < , and | − |+| − | < we have | ( , )− ( , )| < . A variation
on the Arzelà-Ascoli theorem then shows that a subsequence of ( ) ∈N converges point-
wise to a continuous function ∶ R × [0, ∞) → R. We set Ω ≔ { ∈ R} and ≔ |Ω .
The convergence is locally uniform on Ω.
To be approximating the sequence needs to satisfy (⋅, 0) → 0 pointwise, where 0 is
the above extension of the initial function such that (⋅, 0) = 0 holds. Furthermore, for
any , ∈ R and any 0 > 0, we suppose that there is an index = ( , , 0 ) such that
is a smooth solution of (2.1) on the set {| | < , | | < , 0 < | | < }. Moreover,
we assume uniform estimates of the form | | ≤ ( , , , 0 ) for ≥ and for all
multi-indices on this set. The subsequential convergence → is then locally smooth
on Ω ∖ (Ω0 × {0}). As a consequence, solves (2.1) on Ω ∖ (Ω0 × {0}) and is as asserted.
We have summarized how one obtains a solution from an approximating sequence and
what are sufficient conditions on this sequence. One still has to find the approximating
sequence and prove the local estimates on the functions and its derivatives. For the approx-
imations one could either solve initial boundary value problems or use the flow of closed
hypersurfaces which have graphical parts. These are just two options and one cannot say
in general how to approximate; it depends on the given problem.
For the mean curvature flow, we can find an approximating sequence in the following
way. One considers for ∈ R+ the functions ∶ R → R with
⎧ | | < ,
{
( ) ≔ ⎨ ≥ , (2.2)
{− ≤ − .
⎩
62. Mean Curvature Flow Without Singularities Then one mollifies ∘ and restricts to a ball. Solving graphical mean curvature flow on this ball with this initial function and holding the boundary values fixed over time, we find an approximator. (It can be extended to all of R by an arbitrary value.) An approximating sequence is obtained by taking increasingly larger , finer mollification parameter, and larger balls. For the local estimates it is often possible to use the height function to construct a cut-off function. We don’t say anything about the shadow flow here because the details of that part of the proof are not important to us. 7
3. Vain Mean Curvature Flow It is well-known that singularities occur in the mean curvature flow of closed hypersurfaces. For graphical hypersurfaces, however, the situation is much better, as we have already seen in Chapter 2, and no singularities occur. In the present chapter, we investigate a situation where these regimes get in touch. Consider a mirror-symmetric closed hypersurface, i.e., a hypersurface that is symmetric with respect to some hyperplane. Furthermore, assume that the two symmetric parts are graphical over that hyperplane. Since the hypersurface is closed, singularities inevitably arise for the mean curvature flow. But due to the graphical properties, these occur only on the hyperplane of symmetry. Or at least, we show that there exists a weak solution with this behavior. To handle this situation, the notion of vanity has been invented and is described below. 3.1. Vanity Notation 3.1. For ( 1 , …, ) ∈ R we denote with the reflection of in the - direction. So is given by ≔ ( 1 , …, −1 , − ) . (3.1) Moreover, we will write ≔ (1 − ) + for ∈ [0, 1] . (3.2) Definition 3.2. (i) A subset Ω ⊂ R is called vain if all of its points can see their mirror image, i.e., ∀ ∈ Ω ∀ ∈ [0, 1] ∶ ∈ Ω holds. (3.3) (ii) If Ω ⊂ R is vain, then ∶ Ω → R is called vain if ∀ ∈ Ω ∀ ∈ [0, 1] ∶ ( ) ≤ ( ) holds. (3.4) Remark 3.3. (i) By inserting = 1, one can see that any vain set is mirror-symmetrical. (ii) For a vain function ∶ Ω → R holds ( ) = ( ) for all ∈ Ω. (iii) If Ω is vain, then the fibers of the projection Ω ∋ ↦ ( 1 , …, −1 ) are convex, i.e., they are lines. 8
3. Vain Mean Curvature Flow
(iv) A function ∶ Ω → R is vain if and only if the set
{( 1 , …, +1 ) ∈ Ω × R ∶ ( 1 , …, ) ≤ +1 }
is vain in the direction of .
Proposition 3.4. If ∶ Ω → R is vain, then the sets −1 ([−∞, ]) and −1 ([−∞, )) are
vain for any ∈ R.
Proof. Obvious from the definitions.
Proposition 3.5. If is a vain function and is a monotonically increasing function,
then ∘ is vain.
Proof. The inequality ( ) ≤ ( ) immediately implies ∘ ( ) ≤ ∘ ( ).
Proposition 3.6. For two vain functions , ∶ Ω → R, their sum + is vain.
Proposition 3.7. Let Ω be a vain set. A function ∶ Ω → R is vain if and only if
is mirror-symmetrical and for any ( 1 , …, −1 ) ∈ R −1 , ↦ ( 1 , …, −1 , ) is
monotonically increasing on { ∶ ( 1 , …, −1 , ) ∈ Ω, ≥ 0}.
Proof. Firstly, let be vain. Then is mirror-symmetrical (Remark 3.3 (ii)). Let
( 1 , …, −1 ) ∈ R −1 . Let 0 ≤ ≤ be such that ( ) ≔ ( 1 , …, −1 , ) ∈ Ω, and
( )
consequently ( ) ≔ ( 1 , …, −1 , ) ∈ Ω. With ≔ − 2 we can write
( )
= .
Hence, by the vanity of , ( ( ) ) ≤ ( ( ) ) holds, which proves the monotonicity of
↦ ( 1 , …, −1 , ) on the set { ∶ ( 1 , …, −1 , ) ∈ Ω, ≥ 0}.
Now we assume the symmetry and the monotonicity property and prove that is vain.
Let ∈ Ω and ∈ [0, 1]. We shall prove ( ) ≤ ( ). Symmetry is the reason why we
only need to consider ≥ 0 and ≤ 12 . Then ( ) ≤ ( ) follows from the monotonicity
property.
Corollary 3.8. Let ∈ 1 (Ω) be a mirror-symmetrical function. Then is vain if and
only if ( ) ≥ 0 for > 0 holds.
Proposition 3.9. Let Ω be an open, vain set. Then it is of the form
Ω = {( ,̂ ) ∈ R ∶ ̂ ∈ , | | < ℎ( )}
̂ (3.5)
for a function ℎ defined on an open set ⊂ R −1 .
Proof. Let be the projection of Ω to R −1 , where we use the projection to the first
− 1 coordinates. Because Ω is open and vain, the fibers of that projection are lines of
the form
−1 ( )̂ = { }̂ × (−ℎ( ),
̂ ℎ( ))
̂ . (3.6)
Defining ℎ on in this way shows that Ω has the asserted form.
Remark 3.10. For any continuous function ℎ on an open subset ⊂ R −1 , we can define
an open, vain set via (3.5).
Lemma 3.11. For a vain set Ω ⊂ R the negative distance function − ≔ − dist Ω ,
defined on Ω, is vain.
Proof. Let ∈ Ω and ∈ [0, 1] be given. Because Ω is mirror-symmetrical, we have
( ) = ( ). For ∈ R with | | < ( ) = ( ), there hold + ∈ Ω and + ∈ Ω.
Noting Remark 3.3 (iii), we deduce + ∈ Ω. Because was an arbitrary vector subject
to the condition | | < ( ), it follows ( ) ≥ ( ).
93. Vain Mean Curvature Flow
Mollification of Vain Functions. We check whether the standard mollification by con-
volution with a mollifying kernel preserves vanity. For special kernels we can give an
affirmative answer. However, this is not expected for general kernels. (For instance, have
a look at the vain function √| | and the “kernel” 12 ( + − ). Then the convolution is
not vain.)
Let ∈ ∞ (R ) be a Friedrichs mollifier such that − is vain. By Corollary 3.8 this is
equivalent to the conditions ( ) = ( ) for all ∈ R and ( ) ≥ 0 for < 0. For
> 0 we define ( ) ≔ − ( / ). Clearly, this function has these same properties as
does.
Let ∶ R → R be a vain function. The mollification of is defined by ( 0 ) ≔
∫ ( ) ( 0 − ) d . We check ( 0 ) = ( 0 ) and ( 0 ) ≥ 0 for 0 > 0 (we use
= and = + 2 0 (no sum over )):
( 0 ) = ∫ ( ) ( 0 − ) d = ∫ ( ) ( 0 − ) d
(3.7)
= ∫ ( ) ( 0 − ) d = ( 0 ) ,
( 0 ) = ∫ ( ) ( 0 − ) d
= ∫ ( ) ( 0 − ) d + ∫ ( ) ( 0 − ) d
{
0}
= ∫ ( + 2 0 ) (
⏟⏟0−
− 2 ⏟⏟
⏟⏟⏟ 0 ) d
0 − (3.8)
{ >
0}
+ ∫ ( ) ( 0 − ) d
{ >
0}
= ∫ ( ( ) − ( + 2 0 )) ( 0 − ) .
{ >
0}
We have used ( 0 − ) = − ( 0 − ) and have renamed the integration variable
from to in the last step. For 0 < 0 < we have − < + 2 0 < and
( 0 − ) ≥ 0. We deduce from the vanity of that ( ) ≥ ( + 2 0 ) and
therefore obtain from (3.8)
( 0 ) ≥ 0 for 0 > 0 . (3.9)
Together with (3.7), the symmetry of , the vanity of the mollification now follows
from Corollary 3.8.
3.2. Vain Mean Curvature Flow
Proposition 3.12. Let Ω ⊂ R be a vain, open, bounded, and smooth set. Let ∈
2;1 (Ω × [0, )) be a solution of graphical mean curvature flow with ≤ ( ∈ R) and
( , ) = for ∈ Ω, ∈ [0, ). If (⋅, 0) is vain, then (⋅, ) is vain for all ∈ [0, ).
103. Vain Mean Curvature Flow
Proof. Because of the uniqueness of the solution, (⋅, ) is mirror-symmetrical for all ∈
[0, ).
Let (⋅, ) be the downwards pointing normal to ≔ graph (⋅, ) ⊂ R +1 . If we track
points on in time along the normal direction, then on holds (cf. B.17)
− Δ = | |2 , (3.10)
where | |2 is the squared norm of the second fundamental form and Δ denotes the Laplace-
Beltrami operator of . Accordingly, for ≔ = ⟨ , ⟩ ≡ √1+|∇ | 2
holds
− Δ = | |2 . (3.11)
Let + ≔ { ∈ ∶ > 0}. By the mirror-symmetry of (⋅, ), we have (⋅, ) = 0
on the part of + which lies on { = 0}. The vanity of Ω and (⋅, ) ≤ as well as
( , ) = for ∈ Ω imply that (⋅, ) ≥ 0 holds on the remaining part of + (cf.
Proposition 3.9). Furthermore, the vanity of (⋅, 0) implies that (⋅, 0) ≥ 0 on 0+ (cf.
Corollary 3.8).
The maximum principle yields ( , ) ≥ 0 for all ∈ + and for all ∈ [0, ). This in
turn implies the vanity of (⋅, ) for all ∈ [0, ) (Corollary 3.8).
Theorem 3.13. Let Ω0 ⊂ R be an open, vain set and let 0 ∶ Ω0 → R be a vain, locally
Lipschitz, proper function bounded from below such that ( 0 , Ω0 ) forms initial data for a
mean curvature flow without singularities as described in chapter 2. Then there exists a
mean curvature flow without singularities ( , Ω) with these initial data and such that Ω is
a vain set and (⋅, ) is a vain function for all ≥ 0.
Sketch of proof. One uses the construction for the mean curvature flow without singulari-
ties outlined in chapter 2 and checks whether the vanity is preserved in those steps.
• Proposition 3.5 with the monotonical function (2.2) ensures that ∘ 0 is vain. The
properness of 0 together with the boundedness from below ensures that ∘ 0 is
constantly equal to outside a ball.
• For the mollifications: One does these with a mollifier as described above to make
sure that the mollification doesn’t destroy the vanity. The mollifications still have
the property of being constantly equal to outside large balls.
• Proposition 3.12 ensures that the approximating solutions are vain if we take large
enough balls on which we consider the Dirichlet problems. Since the property of
being vain is closed under pointwise limits, the constructed solution is vain.
Because > −∞ by the boundedness from below, we have Ω = −1 ((−∞, ∞)) =
−1 ([−∞, ∞)). Therefore, the vanity of Ω and thus the vanity of the time slices Ω follows
from Proposition 3.4.
Corollary 3.14. Let Ω0 ⊂ R be open and vain. Then there is a weak solution (Ω ) ∈[0,∞)
of the mean curvature flow in the sense of a domain flow (Definition C.4) that starts with
Ω0 and such that Ω is vain for all ≥ 0.
113. Vain Mean Curvature Flow
Proof. Without loss of generality we assume Ω0 ≠ ∅. Let ≔ dist Ω0 be the positive
1
distance function to the boundary on Ω0 . We set 0 ( ) = ( ) +| |2 for ∈ Ω0 . By Lemma
3.11 and Propositions 3.5 and 3.6, 0 is a vain function and 0 satisfies the hypothesis of
Theorem 3.13. Let ∶ Ω → R be a solution from this Theorem (uniqueness of the solution
is not proven). Then the Ω are vain and they form a domain flow.
Theorem 3.15. Let Ω0 ⊂ R be an open and vain set. Suppose Ω0 is bounded in the
-direction, 0 ≔ Ω0 ∩ { > 0} is of class 2 , and suppose that the curvature of 0
is bounded on sets { > } and that is positively bounded from below on these sets.
(The bounds may depend on . is the outwards/upwards pointing normal to 0 .)
Then, for the domain flow (Ω ) ∈[0,∞) from Corollary 3.14, the ≔ Ω ∩ { > 0}
are smooth submanifolds for > 0.
Proof. Let ∶ Ω → R be the mean curvature flow without singularities from the proof of
Corollary 3.14. Let ≔ graph (⋅, ). We will prove uniform estimates for in the
→∞
region ≥ . From these we infer that − +1 −−−→ × R converges locally
smoothly with locally uniform estimates. In particular, is smooth.
For the estimates one would like to use the cut-off function ( − )+ . However, this
function doesn’t cut off compact subsets from the mean curvature flow . For this reason,
one considers cut-off functions similar to those in [EH91] and whose supports are given by
shrinking balls. One chooses larger and larger balls such that in the limit the half-space
> is obtained. More precisely, we do the following construction. For 0 > 0 and
> 0, we set 0 ≔ (0, …, 0, 0 + , 0) ∈ R +1 . We define the corresponding cut-off
function ∶ R +1 × [0, ∞) → R by
1
( , ) ≔ ( 2 − 2 − | − 0 |2 )+ . (3.12)
2 0 0
The support of is given by a shrinking ball around 0 of initial radius 0 . If ( , ) is
fixed and 0 → ∞, in the limit one obtains the cut-off function
1
lim ( , ) = lim (−2 − ∑ ( )2 + 02 − ( − ( 0 + ))2 )
0 →∞ 0 →∞ 2 0
≠
+
1
= lim ( ( 0 − ( − ( 0 + )))( 0 + ( − ( 0 + )))) (3.13)
0 →∞ 2 0
+
1
= lim ( (2 0 − + ) ( − ))
0 →∞ 2 0 +
= ( − )+ .
d
Considering the operator d − Δ on the surface and where > 0, the function
satisfies, with a local parametrization ↦ ( , ) of such that points in normal
123. Vain Mean Curvature Flow
direction,
d 1 d
( − Δ) ( ( , ), ) = (−2 − ( − Δ) | ( , )|2 )
d 2 0 d
1 d
= (−2 − 2 ⟨( − Δ) , ⟩ + 2 |∇ |2 ) (3.14)
2 0 d
1
= (−2 + 2 ) = 0 .
2 0
d
Here we have used ( d − Δ) = 0 and |∇ |2 = ∇ ∇ = = .
In a region where > holds ( > 0), we have for ≔ −
2 2
d ( d − Δ) ∇ ∇
( − Δ) log = d +∣ ∣ =− +∣ ∣ . (3.15)
d
At first, we estimate = , where is the downwards pointing normal to . By the
vanity of , we have ≥ 0 in the region ≥ 0. The strong maximum principle implies
that > 0 in the region > 0 (note (3.11)).
In an interior maximum point of −1 , and consequently of − log + log , there hold
∇ ∇
= and
d
0≤( − Δ) (− log + log )
d
2 2
2 ∇ ∇
= −| | − ∣ ∣ − +∣ ∣ (note (3.11) and (3.15)) (3.16)
= −| |2 − < 0 .
Contradiction. So there can’t be an interior maximum point, and because vanishes on
the lateral boundary of the region { > }, −1 is bounded by the supremum of its
initial values. With → 0 it follows that −1 is bounded by its initial values, too.
Finally, we let 0 → ∞ and obtain the estimate
−1 ( − )+ ≤ sup −1 ( − )+ . (3.17)
=0
| |
For the curvature estimate we consider the test function ≔ − with = 21 inf ,
where the infimum is taken over the set supp . Using (3.17) with ̃ = 2 , we see that
≥ > 0. In an interior maximum point of , there holds (note (B.36), (3.11), (3.14),
and ∣∇| |2 ∣ = |2⟨ , ∇ ⟩| ≤ 2 | | |∇ |)
d d
0≤( − Δ) log 2 = ( − Δ) (log | |2 − 2 log( − ) + 2 log )
d d
2 2 2
|∇ |2 | |4 ∇| |2 ∇ ∇
= −2 2
+2 2
+ ∣ 2
∣ −2 | |2 − 2 ∣ ∣ + 2∣ ∣
⏟⏟⏟ | |
⏟⏟ | | | | − −
(3.18)
1 2 2
≤− 2 ∣ ∇| |
| |2
∣
2 2 2 2
1 ∇| | 2 ∇ ∇
≤ ∣ 2
∣ − | |2 − 2 ∣ ∣ + 2∣ ∣ .
2 | | − −
133. Vain Mean Curvature Flow
For arbitrary > 0 we deduce from the condition ∇ log 2 = 0 at the maximal point
2 2 2 2
∇| |2 ∇ ∇ ∇ ∇
∣ 2
∣ = ∣2 −2 ∣ ≤ 4 (1 + ) ∣ ∣ + 4 (1 + −1 ) ∣ ∣ . (3.19)
| | − −
Therefore,
2 2
2 ∇ ∇
0≤− | |2 + 2 ∣ ∣ + 2 (2 + −1 ) ∣ ∣ . (3.20)
− −
There hold
2
2 − 0
|∇ | = ∣⟨ , ∇ ⟩∣ ≤ 1 ⋅ |∇ |2 = ∇ ∇ = = (3.21)
0
and
|∇ |2 = |∇ |2 = (ℎ ∇ ) (ℎ ∇ ) ≤ | |2 |∇ |2 ≤ | |2 . (3.22)
Substituting these two inequalities into (3.20) yields
−2 2
0≤( + 2
) | |2 + 2 (2 + −1 ) 2 . (3.23)
− ( − )
With the choice = 21 2 , we obtain
−2 2 −2 ( − ) + 2 −2 2 + 2 2
( + ) = ≤ = − . (3.24)
− ( − )2 ( − )2 ( − )2 ( − )2
We conclude
| |2 2
2
≤ 2 (2 + 2 −2 ) −2 ≡ ( , ) . (3.25)
( − )
So, in an interior maximum point, is bounded by a controlled constant. In particular, it
follows that | | ≤ ( , ) (1 + sup =0 (| | )) . With 0 → ∞ we obtain the estimate
| | ( − )+ ≤ (sup (| | ( − )+ ) , sup −1 , ) . (3.26)
=0 >
Together with the estimate (3.17) on , we obtain the curvature estimate
sup | | ≤ ⎛
⎜
⎜ , sup | |, sup −1 , sup , ⎞
⎟
⎟. (3.27)
>2 =0, > 1 =0
⎝ =0, > 2 ⎠
The higher order estimates, i.e., estimates on |∇ |, are omitted because these can be
obtained by following [EH91] from hereon.
We have proven uniform estimates for = graph (⋅, ) for all in regions { ≥ }.
By Proposition 3.9, ∩ { > 0} is graphical over the hyperplane orthogonal to . Let
ℎ be the representing function. Then ℎ is bounded and ℎ is monotonically increasing in
the +1 direction. The inequality (3.17) yields a gradient estimate on ℎ where ℎ > . The
estimates on the second fundamental form (3.27) and on its higher derivatives provide us
with estimates on the higher derivatives of ℎ. We conclude that there is a smooth limit
of ℎ(⋅, +1 ) for +1 → ∞. This limit is a graphical representation of . Hence, is
smooth.
143. Vain Mean Curvature Flow
Remark 3.16. As a byproduct, Theorem 3.15 (see also Proposition 3.9) provides a solution
of a free boundary value problem where the boundary moves on a hyperplane and the
hypersurface meets that hyperplane perpendicularly. Namely, the family ( ) ∈[0,∞) is a
family of smooth graphical hypersurfaces over a hyperplane that solves the mean curvature
flow ( may be empty). The boundaries , which reside on the hyperplane, may be
singular, however. But at spacetime-points (on the hyperplane) where Ω is smooth
is smooth too and by symmetry of Ω the normal to Ω lies in the hyperplane such
that = Ω ∩ { > 0} meets the hyperplane perpendicularly. In this way can be
viewed as a smooth graphical solution to the free Neumann boundary value problem with
singularities at the boundary.
As was pointed out to the author by O. Schnürer, it is not clear that always meets
the hyperplane perpendicularly. To explain this, one needs to think about singularities of
Ω where it is possible to continuously extend the normal coming from one of the two
sides but where the limits from the two sides disagree. In this case it may be possible that
the normal for is definable on the hyperplane but that it points out of the hyperplane.
Figure 3.1.: The free Neumann boundary value problem from Remark 3.16. Depicted is a
solution at two times, one before and one after a singularity which appeared
at the boundary on the plane.
154. Barrier Over an Annulus In the present chapter we assume ≥ 2. Central to this chapter is a barrier which is defined over annuli. This barrier enables us to prove estimates for a mean curvature flow in terms of the height over an annulus earlier in time. This will be exploited to obtain various results. Figure 4.1.: Sketch of the barrier. The upper barrier has the following geometry (Fig. 4.1). Initially start with an annulus in -dimensional Euclidean space. Over this, we consider a special function which tends to infinity at the boundary of the annulus. As time goes by, the initial annulus shrinks by mean curvature flow and the function is adjusted accordingly to have the shrinking annulus as its domain; while at the same time, the function is shifted upwards. In the first section, we will write down the barrier and prove that it really functions as a barrier. The upshot is Theorem 4.2. Afterwards, in the second section, we will utilize the barrier to construct an “ugly” example of mean curvature flow without singularities with wild oscillations which persist up to the vanishing time. As another application of the barrier, we will prove in the third section a certain relationship between the spatial asymptotics of a mean curvature flow without singularities and its temporal asymptotics at the vanishing time. 4.1. Construction of the Barrier The barrier construction starts with a hypersurface that is obtained from a grim reaper curve which is rotated around the +1 -axis. The grim reaper is a well-known special solution of curve-shortening flow (one-dimensional mean curvature flow). In that context it is an important tool and can be helpful in diverse situations and it is often times used as a barrier. The grim reaper has the explicit graphical representation ( > 0) 1 gr ( , ) = − log cos( ) + for ∈ (− , ), ∈ R . (4.1) 2 2 In this form graph gr translates with speed upwards. 16
4. Barrier Over an Annulus
Our barrier construction is motivated by this solution and initially we start with a grim
reaper curve rotated around a vertical axis. Let us have a look at the barrier function
now:
1
( , ) = − log cos [ (√| |2 + 2( − 1) − )] + + ( ) , (4.2)
where , > 0. As can be seen, is similar to gr . But is replaced by the term
√| |2 + 2( − 1) which is motivated by the radius of an ( − 1)-dimensional sphere
flowing by its mean curvature. The function ( ) will be adequately chosen to make
satisfy the correct differential inequality. We will give two possible choices for , one
rather simple, the other more elaborate but much smaller as becomes large. However,
we include the second one only for the sake of completeness because in our applications
dominates ( ) for large in either case.
Before we start proving the differential inequality, we must talk about the domain of
definition of . We set
2
∈ ( ) ≔ [0, ( )] ≔ [0, ] (4.3)
2( − 1)
∈ ( , ) ≔ { ∈ R ∶ ( − ) < √| |2 + 2( − 1) < ( + )} . (4.4)
2 2
( , ) is an annulus or a ball, depending on , , and .
Differential
√ Inequality. To keep the computations √ accessible, we will make the abbrevi-
ations 2
for √| | + 2( − 1) and [ ⋅ ] for ( − ).
To show that is an upper barrier, we must prove
!
− ̇ + ( − ) ≤ 0. (4.5)
1 + |∇ |2
First, we determine the derivatives.
−1
̇ = tan[ ⋅ ] √ + + ′ ( ) , (4.6)
= tan[ ⋅ ] √ , (4.7)
tan[ ⋅ ]
= (1 + tan2 [ ⋅ ]) √ 2 + √ ( − √ 2 ) . (4.8)
174. Barrier Over an Annulus Substituting into (4.5) yields − ̇ + ( − ) 1 + |∇ |2 1 = − ̇ + ((1 + |∇ |2 ) − ) 1 + |∇ |2 −1 = − − ′ ( ) − tan[ ⋅ ] √ (1 + tan2 [ ⋅ ]) | |2 | |2 | |4 + 2 ((1 + tan2 [ ⋅ ] √ 2 ) √ 2 − tan2 [ ⋅ ] √ 4 ) 1 + tan2 [ ⋅ ] √| | 2 tan[ √ ⋅] | |2 2 | |2 + 2 ((1 + tan [ ⋅ ] √ 2 ) ( − √ 2 ) 1 + tan2 [ ⋅ ] √| | 2 (4.9) 2 | |2 | |4 − tan [ ⋅ ] ( √ 2 − √ 4 )) tan[ ⋅ ] (1 + tan2 [ ⋅ ]) | |2 = − − ′ ( ) − √ ( − 1) + 2 ( √ 2) 1 + tan2 [ ⋅ ] √| | 2 tan[ √ ⋅] | |2 | |2 + | |2 ((1 + tan2 [ ⋅ ] √ 2 ) ( − 1) + (1 − √ 2 )) 1 + tan2 [ ⋅ ] √ 2 | |2 (1 − √ 2) tan[ ⋅ ] ! = − ′ ( ) + 2 ( √ − ) ≤ 0 . 1 + tan2 [ ⋅ ] √| | 2 tan[ √ ⋅] In the case ≤ , the inequality clearly holds. Thus, let us from now on assume tan[ √ ⋅]> . Neglecting terms with the appropriate signs, we aim to show tan[ √ ⋅] ! 2 ≤ ′ ( ) . (4.10) tan [ ⋅ ] 2 1+ √ 2 | | If we can choose in such a way then the differential inequality (4.5) follows. We are going to utilize the following lemma. Lemma 4.1. Let ⊂ R>0 be an interval and ∶ → R>0 a monotonically increasing function, which is continuous and surjective. Then ( ) 1 2 2 ≤ (4.11) 1 + ( ) holds for all ∈ , where is the unique solution of ( ) = 1 . 18
4. Barrier Over an Annulus 1 Proof. It is not hard to see that there is a unique solution of ( ) = given the hypothesis on . We distinguish two cases. If ≤ holds, we find ( ) 1 2 2 ≤ ( ) ≤ ( ) = . (4.12) 1 + ( ) In the case that > holds, we have ( ) ( ) 2 2 ≤ . (4.13) 1 + ( ) 1 + 2 ( ) 2 The expression 1+ 2 2 tends to 0 for both → 0 and → ∞. Hence, it attains its maximum at an interior point . We can determine from the extremality condition d 1 + 2 2 − 2 2 2 1 − 2 2 0= ( )∣ = = . (4.14) d 1 + 2 2 = (1 + 2 2 )2 (1 + 2 2 )2 1 We infer that = is the maximum point. That leads to 1/ 1 2 2 ≤ 2 2 = < (4.15) 1+ 1+ 1+1 for any ∈ R>0 . Bringing together (4.13) and (4.15), the assertion follows. Application of the Lemma. We shall derive (4.10) with the help of Lemma 4.1. Clearly, tan[ √ ⋅ ] , seen as a function of ≔ | |, will take the role of ( ). Notice that we fix ∈ ( ) here. The interval is appropriately chosen to be π 2 = (√ 2 − 2( − 1) , √( + ) − 2( − 1) ) , (4.16) 2 √ such that [ ⋅ ] ≔ ( − ) ∈ (0, 2π ). This makes ( ) = tan[ √ ⋅ ] well defined on and surjective onto R>0 . Now we √ show, that our choice of is monotonically increasing. The derivative with respect to is √ √ d tan[ ( − )] tan[ ⋅ ] − sin[ ⋅ ] cos[ ⋅ ] √ √ =√ 2 − √ 2 = √ 2 d cos [ ⋅ ] cos2 [ ⋅ ] √ (4.17) −[⋅] ≥√ 2 =√ 2 >0. cos2 [ ⋅ ] cos2 [ ⋅ ] √ √ So tan[ √ ⋅ ] is increasing as a function of . And because is increasing in , ( ) = tan[ √ ⋅]is increasing in as well. Finally, Lemma 4.1 is applicable and yields tan[ √ ⋅] 1 2 ≤ , (4.18) 1+ tan [ ⋅ ] √ 2 | |2 19
4. Barrier Over an Annulus where is the unique solution in of the equation tan [ (√ 2 + 2( − 1) − )] 1 = . (4.19) √ 2 + 2( − 1) To further estimate 1 in (4.18), we must extract information from (4.19). We will now demonstrate two possible ways to do so. First choice for . We notice that √ 2 + 2( − 1) π tan ( (√ 2 + 2( − 1) − )) = ≥ 1 = tan ( ) . (4.20) 4 Consequently, and by 2( − 1) ≤ 2 , π 2 π 2 ≥ ( + ) − 2( − 1) ≥ . (4.21) 4 2 Choosing 2 ( ) ≔ √ (4.22) π we infer from (4.18) and (4.21) tan[ √ ⋅] 1 2 ≤ ≤√ = ′ ( ) . (4.23) 1+ 2 tan [ ⋅ ] √ 2 | |2 π I.e., (4.10), and hence (4.5) hold. Second choice for (optional). We write ≔ √ 2 + 2( − 1) . (4.24) π π We note that depends on and lies in the range [ + 4 , + 2 ), which can be seen from (4.20). With at hand we can rewrite (4.19) as 1 1 tan( ( − )) = = . (4.25) √ 2 − 2( − 1) This can easily be solved for 2( − 1) . We write ( ) for 2( − 1) viewed as a function of : 1 π π ( ) ≔ 2 (1 − 2 ), ∈ [ + 4 , + 2 ) . (4.26) tan ( ( − )) π π 2 We continuously extend to the closed interval through ( + 2 ) = ( + 2 ) . We will estimate ( ) from below by π ( ) ≔ (1 − 4 ( + )) 2 + 4 3 ; (4.27) 2 20
4. Barrier Over an Annulus
π π π 2
is the solution of ′ ( ) = 2 ( ) + 4 2 with ( + 2 ) = ( + 2 ) = ( + 2 ) . The
function fulfills the differential inequality
2
1 2 1 + tan ( ( − ))
′ ( ) = 2 (1 − ) + 2
tan2 ( ( − )) tan3 ( ( − )) (4.28)
2
≤ ( ) + 4 2 .
It follows that
d 2 2 2
( ( ) − ( )) ≤ ( ) + 4 2 − ( ( ) + 4 2 ) = ( ( ) − ( )) . (4.29)
d
π
In fact the inequality is strict except for = + 4 . Thus, whenever ( ) = ( ) for
π π ′ ′
some ∈ ( + 4 , + 2 ), we have ( ) < ( ) and there is an > 0 such that >
on ( − , ). Considering sup{ ∶ ( ) < ( )}, it is now easy to show that ≥ on the
π π π π
whole interval [ + 4 , + 2 ]. So we have shown for ∈ [ + 4 , + 2 ]
π
)) 2 + 4 3 ,
( ) ≥ (1 − 4 ( + (4.30)
2
( ) ( ) π
≥ 2 ≥ 1 − 4 ( + ) + 4 , (4.31)
π 2 2
( + 4 )
π 1 ( )
≤ + − (1 − )
2 4 π 2
( + 4 )
(4.32)
1 π 1 ( )
= + ( − (1 − )) .
2 4 π 2
( + 4 )
Let us return to (4.24), the value for we are interested in. Let us also remember that
π
( ) = 2( − 1) holds in this context. With ≥ + 4 and (4.32) we can continue from
(4.25)
1 tan( ( − )) 1 π 1 2( − 1)
≤ ≤ π tan ( − (1 − 2 )) . (4.33)
+ 4 2 4 ( + π ) 4
We integrate the last term:
π
2 ( + 4 ) 1 2( − 1) 1
( ) ≔ [− log sin ( (1 − 2 )) + log sin ( 4 )] . (4.34)
−1 4 π
( + 4 )
Then we obtain (4.10) from (4.33) and (4.18). So the differential inequality (4.5) for
follows with this choice for .
Asymptotics of for large (optional). The merit √ of the second choice for is the
milder growth in . While (4.22) clearly grows like , (4.34) only grows like log( ). To
see this let us evaluate at the final time ( ) of the time interval under consideration
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