the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

the quermassintegral inequalities for k convex starshaped domains abstract we give a proof of the isoperimetric inequality for quermassintegrals of non convex starshaped domains using a result of gerhardt c gerhardt flow of nonconvex hypersurfaces into spheres j differential geometry 32 1990 299 314 and urbas j urbas on the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures math z 205 1990 355 372 on an expanding geometric curvature flow

universality in mean curvature flow neckpinches we study noncompact surfaces evolving by mean curvature flow without any symmetry assumptions we prove that any solution that is c 3 close at some time to a standard neck will develop a neckpinch singularity in finite time will become asymptotically rotationally symmetric in a space time neighborhood of its singular set and will have a unique tangent flow

generic mean curvature flow i generic singularities it has long been conjectured that starting at a generic smooth closed embedded surface in r 3 the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders that is the only singularities of a generic ow are spherical or cylindrical we will address this conjecture here and in a sequel the higher dimensional case will be addressed elsewhere the key to showing this conjecture is to show that shrinking spheres cylinders and planes are the only stable self shrinkers under the mean curvature ow we prove this here in all dimensions an easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away

the singular set of mean curvature flow with generic singularities a mean curvature flow starting from a closed embedded hypersurface in mathbf r n 1 must develop singularities we show that if the flow has only generic singularities then the space time singular set is contained in finitely many compact embedded n 1 dimensional lipschitz submanifolds plus a set of dimension at most n 2 if the initial hypersurface is mean convex then all singularities are generic and the results apply in mathbf r 3 and mathbf r 4 we show that for almost all times the evolving hypersurface is completely smooth and any connected component of the singular set is entirely contained in a time slice for 2 or 3 convex hypersurfaces in all dimensions the same arguments lead to the same conclusion the flow is completely smooth at almost all times and connected components of the singular set are contained in time slices a key technical point is a strong parabolic reifenberg property that we show in all dimensions and for all flows with only generic singularities we also show that the entire flow clears out very rapidly after a generic singularity these results are essentially optimal

on the type iib solutions to mean curvature flow abstract in this paper we study the type iib mean curvature flow we first prove that if the convex entire graph y u y over r n n 2 satisfying there exist positive constants c and n such that u r c r for r n the longtime solution to mean curvature flow with initial data y u y must be type iib we also study the asymptotic behavior of type iib mean curvature flow and show that the limit of suitable rescaling sequence for mean convex type iib mean curvature flow satisfying andrews noncollapsing condition is translating soliton

uniqueness of weak solutions to a high dimensional keller segel equation with degenerate diffusion and nonlocal aggregation abstract this paper considers weak solutions to the degenerate keller segel equation with nonlocal aggregation u t u m u b u in r d r where b u 2 u d 3 2 d 1 m 2 d in a previous paper of the authors hong et al 2015 a criterion was established for global existence versus finite time blow up of weak solutions to the problem a natural question is whether the uniqueness is true for the weak solutions obtained a positive answer is given in this paper that the global weak solutions must be unique provided the second moment of initial data is finite which means that the weak solutions are weak entropy solutions in fact the framework of the proof is based on the optimal transportation method

convexity estimates for a nonhomogeneous mean curvature flow we study the evolution of a closed immersed hypersurface whose speed is given by a function phi h h of the mean curvature asymptotic to h ln h for large h compared with other nonlinear functions of the curvatures this speed has some good properties which allow for an easier study of the formation of singularities in the nonconvex case we prove apriori estimates showing that any surface with positive mean curvature at the initial time becomes asymptotically convex near a singularity similar estimates also hold for the mean curvature flow for the flow considered here they admit a simpler proof based only on the maximum principle

a microscopic convexity principle for nonlinear partial differential equations we study microscopic convexity property of fully nonlinear elliptic and parabolic partial differential equations under certain general structure condition we establish that the rank of hessian 2u is of constant rank for any convex solution u of equation f 2u u u x 0 the similar result is also proved for parabolic equations some of geometric applications are also discussed

the extension problem of the mean curvature flow i we show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in mathbb r 3

uniqueness of convex ancient solutions to mean curvature flow in mathbb r 3 r 3 a well known question of perelman concerns the classification of noncompact ancient solutions to the ricci flow in dimension 3 which have positive sectional curvature and are kappa noncollapsed in this paper we solve the analogous problem for mean curvature flow in mathbb r 3 and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in mathbb r 3 which is strictly convex and noncollapsed

uniqueness of blowups and łojasiewicz inequalities once one knows that singularities occur one naturally wonders what the singu larities are like for minimal varieties the first answer already known to federer fleming in 1959 is that they weakly resemble cones 1 for mean curvature flow by the combined work of huisken ilmanen and white singularities weakly resemble shrinkers unfortunately the simple proofs leave open the possibility that a minimal variety or a mean curvature flow looked at under a microscope will resemble one blowup but under higher magnification it might as far as anyone knows resemble a completely different blowup whether this ever happens is perhaps the most fundamental question about singularities it is this long stand ing open question that we settle here for mean curvature flow at all generic singularities and for mean convex mean curvature flow at all singularities

invariant hypersurfaces with linear prescribed mean curvature abstract our aim is to study invariant hypersurfaces immersed in the euclidean space r n 1 whose mean curvature is given as a linear function in the unit sphere s n depending on its gauss map these hypersurfaces are closely related with the theory of manifolds with density since their weighted mean curvature in the sense of gromov is constant in this paper we obtain explicit parametrizations of constant curvature hypersurfaces and also give a classification of rotationally invariant hypersurfaces

crystalline mean curvature flow of convex sets we prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in open image in new window this theorem can handle the facet breaking bending phenomena and can be generalized to any anisotropic mean curvature flow the method provides also a generalized geometric evolution starting from any compact convex set existing up to the extinction time satisfying a comparison principle and defining a continuous semigroup in time we prove that when the initial set is convex our evolution coincides with the flat curvature flow in the sense of almgren taylor wang as a by product it turns out that the flat curvature flow starting from a compact convex set is unique

the size of the singular set in mean curvature flow of mean convex sets in this paper we study the singularities that form when a hypersurface of positive mean curvature moves with a velocity that is equal at each point to the mean curvature of the surface at that point it is most convenient to describe the results in terms of the level set flow also called biggest flow i2 of chen giga goto cgg and evans spruck es under the level set flow any closed set k in r generates a one parameter family of closed sets ft k t 0 with f0 k k if the boundary of k is a smooth compact hypersurface then so is the boundary of ft k for t in some interval 0 and for such t s the evolution coincides with motion by mean curvature as defined classically by partial differential equations and differential geometry as in h1 however if k is compact then the boundary of ft k will necessarily become singular at some finite time our goal is to show that the singular sets are necessarily quite small this we do provided the initial set k is compact and mean convex in the sense that ft k interior k for all t 0 in case m k is a smooth hypersurface we have the following equivalent characterization of mean convexity k is mean convex if and only if the mean curvature of m with respect to the inward unit normal is everywhere non negative given a compact set k let k be the region in spacetime swept out by the ft k k x t r r t 0 x ft k a point x x t in the boundary of k is called a regular point if 1 x has a neighborhood in which k is a smooth manifold with boundary and 2 the tangent plane to k at x is not horizontal i e is not r 0 note that if x x t is a regular point then in a neighborhood of x ft k is a smooth manifold with boundary in r a point x x t in k with t 0 that is not a regular point is called a singular point

inequalities for quermassintegrals on k convex domains abstract in this paper we study the aleksandrov fenchel inequalities for quermassintegrals on a class of non convex domains our proof uses optimal transport maps as a tool to relate curvature quantities of different orders defined on the boundary of the domain

a projective model structure on pro simplicial sheaves and the relative etale homotopy type abstract in this work we shall introduce a new model structure on the category of pro simplicial sheaves which is very convenient for the study of etale homotopy using this model structure we define a pro space associated with a topos as a result of applying a derived functor we show that our construction lifts artin and mazur s etale homotopy type 3 in the relevant special case our definition extends naturally to a relative notion namely a pro object associated with a map of topoi this relative notion lifts the relative etale homotopy type that was used in 18 for the study of obstructions to the existence of rational points this relative notion enables to generalize these homotopical obstructions from fields to general base schemas and general maps of topoi our model structure is constructed using a general theorem that we prove namely we introduce a much weaker structure than a model category which we call a weak fibration category our theorem says that a weak fibration category can be completed into a full model category structure on its pro category provided it satisfies some additional technical requirements our model structure is obtained by applying this result to the weak fibration category of simplicial sheaves over a grothendieck site where the weak equivalences and the fibrations are local in the sense of jardine 23

a fully nonlinear conformal flow on locally conformally flat manifolds we study a fully nonlinear flow for conformal metrics the long time existence and the sequential convergence of flow will be established for locally conformally flat manifolds as an application we solve the k yamabe problem for locally conformal flat manifolds when k n 2

the interior gradient estimate of hessian quotient equations abstract in this paper we establish the interior gradient estimate of k admissible solutions of hessian quotient equations k d 2 u l d 2 u f x u d u with 0 l k n which generalizes the result of 1 7

profinite completion of operads and the grothendieck teichmuller group abstract in this paper we prove that the group of homotopy automorphisms of the profinite completion of the operad of little 2 disks is isomorphic to the profinite grothendieck teichmuller group in particular the absolute galois group of q acts faithfully on the profinite completion of e 2 in the homotopy category of profinite weak operads

comparison of stable homotopy categories and a generalized suslin voevodsky theorem abstract let k be an algebraically closed field of exponential characteristic p given any prime l p we construct a stable etale realization functor et l s pt k pro s pt h z l from the stable category of motivic p 1 spectra over k to the stable category of h z l local pro spectra see section 3 for the definition this is induced by the etale topological realization functor a la friedlander the constant presheaf functor naturally induces the functor sh 1 p sh k 1 p where k and p are as above and sh and sh k are the classical and motivic stable homotopy categories respectively we use the stable etale realization functor to show that this functor is fully faithful furthermore we conclude with a homotopy theoretic generalization of the etale version of the suslin voevodsky theorem

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

the nature of singularities in mean curvature flow of mean convex sets let k be a compact subset of r or more generally of an n 1 dimensional riemannian manifold we suppose that k is mean convex if the boundary of k is smooth and connected this means that the mean curvature of k is everywhere nonnegative with respect to the inward unit normal and is not identically 0 more generally it means that ft k is contained in the interior of k for t 0 where ft k is the set obtained by letting k evolve for time t under the level set mean curvature flow as k evolves it traces out a closed set k of spacetime k x t r r x ft k also there is associated to k a brakke flow m t 7 mt of rectifiable varifolds we call the pair m k a mean convex flow let x x t be a point in spacetime with t 0 suppose xi ti is a sequence of points converging to x and i is a sequence of positive numbers tending to infinity translate the pair m and k in spacetime by y 7 y xi ti and then dilate parabolically by y 7 iy i to get new flows mi and ki the sequence mi ki is called a blow up sequence at x general compactness theorems guarantee that this sequence will have subsequential limits a subsequential limit m k is called a limit flow here m t 7 m t

a fully nonlinear flow for two convex hypersurfaces in riemannian manifolds we consider a one parameter family of closed embedded hypersurfaces moving with normal velocity g kappa big sum i j frac 1 lambda i lambda j 2 kappa big 1 where lambda 1 le cdots le lambda n denote the curvature eigenvalues and kappa is a nonnegative constant this defines a fully nonlinear parabolic equation provided that lambda 1 lambda 2 2 kappa in contrast to mean curvature flow this flow preserves the condition lambda 1 lambda 2 2 kappa in a general ambient manifold our main goal in this paper is to extend the surgery algorithm of huisken sinestrari to this fully nonlinear flow this is the first construction of this kind for a fully nonlinear flow as a corollary we show that a compact riemannian manifold satisfying overline r 1313 overline r 2323 ge 2 kappa 2 with non empty boundary satisfying lambda 1 lambda 2 2 kappa is diffeomorphic to a 1 handlebody the main technical advance is the pointwise curvature derivative estimate the proof of this estimate requires a new argument as the existing techniques for mean curvature flow due to huisken sinestrari haslhofer kleiner and brian white cannot be generalized to the fully nonlinear setting to establish this estimate we employ an induction on scales argument this relies on a combination of several ingredients including the almost convexity estimate the inscribed radius estimate as well as a regularity result for radial graphs we expect that this technique will be useful in other situations as well

self similar solutions and translating solitons for lagrangian mean curvature flow we construct many self similar and translating solitons for lagrangian mean curvature flow including self expanders and translating solitons with arbitrarily small oscillation on the lagrangian angle our translating solitons play the same role as cigar solitons in ricci flow and are important in studying the regularity of lagrangian mean curvature flow given two transverse lagrangian planes wn in cn with sum of characteristic angles less than a we show there exists a lagrangian self expander asymptotic to this pair of planes the maslov class of these self expanders is zero thus they can serve as local models for surgeries on lagrangian mean curvature flow families of self shrinkers and self expanders with different topologies are also constructed this paper generalizes the work of anciaux i joyce 12 lawlor 15 and lee and wang 17 18

convexity estimates for mean curvature flow with free boundary abstract we prove the estimates of 3 and 2 for finite time singularities of mean convex mean curvature flow with free boundary in a barrier s here s can be any embedded oriented surface in r n 1 of bounded geometry and positive inscribed radius we also prove the estimate 4 in the case of convex flows and s s n which gives an alternative proof to 6

contracting convex hypersurfaces in riemannian manifolds by their mean curvature which shrink towards the center of the initial sphere in finite time it was shown in 3 that this behaviour is very typical if the initial hypersurface m o o r 1 is uniformly convex then the surfaces m t contract smoothly to a single point in finite time and the shape of the surfaces becomes spherical at the end of the contraction if the ambient space n is a general riemannian manifold the curvature of n will interfere with the mot ion of the surfaces m r we want to show here that the contract ion first to a small sphere and then to a single point is still

volume growth and bernstein theorems for translating solitons abstract in this paper we study properties of the potential function of a translating soliton m in r n 1 and the volume growth of the intersection of euclidean balls with m we show that if m is a convex translating soliton then the infimum of the mean curvature function on m is zero i e inf m h 0 we give a condition to obtain the bernstein theorem for the translating solitons we also give an outline of a simple proof of the bernstein theorem due to bombieri de giorgi miranda

mean curvature flow of mean convex hypersurfaces in the last 15 years white and huisken sinestrari developed a far reaching structure theory for the mean curvature flow of mean convex hypersurfaces their papers provide a package of estimates and structural results that yield a precise description of singularities and of high curvature regions in a mean convex flow in the present paper we give a new treatment of the theory of mean convex and k convex flows this includes 1 an estimate for derivatives of curvatures 2 a convexity estimate 3 a cylindrical estimate 4 a global convergence theorem 5 a structure theorem for ancient solutions and 6 a partial regularity theorem our new proofs are both more elementary and substantially shorter than the original arguments our estimates are local and universal a key ingredient in our new approach is the new noncollapsing result of andrews 2 some parts are also inspired by the work of perelman 32 33 in a forthcoming paper 17 we will give a new construction of mean curvature flow with surgery based on the methods established in the present paper note added in may 2015 since the first version of this paper was posted on arxiv in april 2013 the estimates have been used to construct mean convex flow with surgery in 3 by brendle and huisken 5 in september 2013 and in another paper by the authors in april 2014 2016 wiley periodicals inc

formation of singularities in the motion of plane curves under hyperbolic mean curvature flow abstract this paper concerns the hyperbolic mean curvature flow hmcf for plane curves a quasilinear wave equation is derived and studied for the motion of plane curves under the hmcf based on this we investigate the formation of singularities in the motion of these curves in particular we prove that the motion under the hmcf of periodic plane curves with small variation on one period and small initial velocity in general blows up and singularities develop in finite time some blowup results have been obtained and the estimates on the life span of the solutions are given

mean curvature flow with surgeries of two convex hypersurfaces we consider a closed smooth hypersurface immersed in euclidean space evolving by mean curvature flow it is well known that the solution exists up to a finite singular time at which the curvature becomes unbounded the purpose of this paper is to define a flow after singularities by a new approach based on a surgery procedure compared with the notions of weak solutions existing in the literature the flow with surgeries has the advantage that it keeps track of the changes of topology of the evolving surface and thus can be applied to classify all geometries that are possible for the initial manifold our construction is inspired by the procedure originally introduced by hamilton for the ricci flow and then employed by perelman in the proof of thurston s geometrization conjecture in this paper we consider initial hypersurfaces which have dimension at least three and are two convex that is such that the sum of the two smallest principal curvatures is nonnegative everywhere under these assumptions we construct a flow with surgeries which has uniformly bounded curvature until the evolving manifold is split in finitely many components with known topology as a corollary we obtain a classification up to diffeomorphism of the hypersurfaces under consideration

mean curvature flow with surgery we give a new proof for the existence of mean curvature flow with surgery of 2 convex hypersurfaces in r as announced in hk13a our proof works for all n 3 including mean convex surfaces in r we also derive a priori estimates for a more general class of flows in a local and flexible setting

a sharp bound for the inscribed radius under mean curvature flow we consider a family of embedded mean convex hypersurfaces which evolve by the mean curvature flow it follows from general results of white that the inscribed radius at each point on the hypersurface is at least frac c h where c is a positive constant that depends only on the initial data andrews recently gave a new proof of that fact using the maximum principle in this paper we show that the inscribed radius is at least frac 1 1 delta h at each point where the curvature is sufficiently large