Felix Schlenk
We consider a connected smooth $n$-dimensional manifold $M$ endowed with a volume form $Ω$, and we show that an open subset $U$ of $R^n$ of Lebesgue measure $\Vol (U)$ embeds into $M$ by a smooth volume preserving embedding whenever the volume condition $\Vol (U) \le \Vol (M,Ω)$ is met.
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We study the long time behavior of the volume preserving $p$-flow in $\mathbb{R}^{n+1}$ for $1\leq p<\frac{n+1}{n-1}$. By extending Andrews' technique for the flow along the affine normal, we prove that every centrally symmetric solution to the volume preserving $p$-flow converges sequentially to the unit ball in the $C^{\infty}$ topology, modulo the group of special linear transformations.
We obtain estimates on nonlocal quantities appearing in the Volume Preserving Mean Curvature Flow (VPMCF) in the closed, Euclidean setting. As a result we demonstrate that blowups of finite time singularities of VPMCF are ancient solutions to Mean Curvature Flow (MCF), prove that monotonicity methods may always be applied at finite times and obtain information on the asymptotics of the flow.
In this paper, we investigate the volume-prserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric space. We prove that the tubeness is preserved along the flow under certain conditions.
We study a volume preserving curvature flow of convex hypersurfaces, driven by a power of the $k$-th elementary symmetric polynomial in the principal curvatures. Unlike most of the previous works on related problems, we do not require assumptions on the curvature pinching of the initial datum. We prove that the solution exists for all times and that the speed remains bounded and converges to a constant in an integral norm. In the case of the volume preserving scalar curvature flow, we can prove that the hypersurfaces converge smoothly and exponentially fast to a round sphere.
We prove: "If $M$ is a compact hypersurface of the hyperbolic space, convex by horospheres and evolving by the volume preserving mean curvature flow, then it flows for all time, convexity by horospheres is preserved and the flow converges, exponentially, to a geodesic sphere". In addition, we show that the same conclusions about long time existence and convergence hold if $M$ is not convex by horospheres but it is close enough to a geodesic sphere.
Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus
本文证明在平坦环面上，从严格稳定临界集附近出发的保体积平均曲率流与表面扩散流全局存在，并指数收敛到该集合的平移。
We study the convergence of an axially symmetric hypersurface evolving by volume preserving mean curvature flow. Assuming the surface is not pinching off along the axis at any time during the flow, and without any additional conditions, as for example on the curvature, we prove that it converges to a hemisphere, when the hypersurface has a free boundary and satisfies Neumann boundary data, and to a sphere when it is compact without boundary.
We show the existence of a weak bi-invariant symmetric nondegenerate 2-form on the volume-preserving diffeomorphism group of a three-dimensional manifold and study its properties. Despite the fact that the space $\mathcal{D}_μ(M^3)$ is infinite-dimensional, we succeed in defining the signature of the bi-invariant quadric form. It is equal to the $η$-invariant of the manifold $M^3$.
We consider a convex Euclidean hypersurface that evolves by a volume or area preserving flow with speed given by a general nonhomogeneous function of the mean curvature. For a broad class of possible speed functions, we show that any closed convex hypersurface converges to a round sphere. The proof is based on the monotonicity of the isoperimetric ratio, which allows to control the inner and outer radius of the hypersurface and to deduce uniform bounds on the curvature by maximum principle arguments.
In this paper, we introduce a definition of $λ$-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean space. We prove that $λ$-hypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete $λ$-hypersurfaces with polynomial area growth and $H-λ\geq 0$, which are generalizations of the results due to Huisken, Colding-Minicozzi. We also define a $\mathcal{F}$-functional and study $\mathcal{F}$-stability of $λ$-hypersurfaces, which extend a result of Colding-Minicozzi. Lower bound growth and…
Let N be a (n+1)-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface. We consider curvature flows in N with different curvature functions F (including the mean curvature, the gauss curvature and the second elementary symmetric polynomial) and a volume preserving term. Under suitable assumptions we prove the long time existence of the flow and the exponential convergence of the corresponding graphs in the $C^\infty$-topology to a hypersurface of constant F-curvature. Furthermore we examine stability properties and foliations of constant F-curvature hypersurfac…
We classify the volume preserving stable hypersurfaces in the real projective space $\mathbb{RP}^n$. As a consequence, the solutions of the isoperimetric problem are tubular neighborhoods of projective subspaces $\mathbb{RP}^k\subset \mathbb{RP}^n$ (starting with points). This confirms a conjecture of Burago and Zalgaller from 1988 and extends to higher dimensions previous result of M. Ritoré and A. Ros on $\mathbb{RP}^3$. We also derive an Willmore type inequality for antipodal invariant hypersurfaces in $\mathbb{S}^n$.
Let (N,g) be a Riemannian manifold. For a compact, connected and oriented submanifold M of N. we define the space of volume preserving embeddings Emb_μ(M,N) as the set of smooth embeddings f:M \rightarrow N such that f*μ^{f}=μ, where μ^{f} (resp. μ) is the Riemannian volume form on f(M) (resp. M) induced by the ambient metric g (the orientation on f(M) being induced by f). In this article, we use the Nash-Moser inverse function Theorem to show that the set of volume preserving embeddings in Emb_μ(M,N) whose mean curvature is nowhere vanishing forms a tame Fréchet manifold, and determine explic…
We study a volume/area preserving curvature flow of hypersurfaces that are convex by horospheres in the hyperbolic space, with velocity given by a generic positive, increasing function of the mean curvature, not necessarly homogeneous. For this class of speeds we prove the exponential convergence to a geodesic sphere. The proof is ispired by [10] and is based on the preserving of the convexity by horospheres that allows to bound the inner and outer radii and to give uniform bounds on the curvature by maximum principle arguments. In order to deduce the exponential trend, we study the behaviour …
Complete bounded $λ$-hypersurfaces in the weighted volume-preserving mean curvature flow
本文研究加权保体积平均曲率流中完备有界 λ-超曲面，建立体积比较定理并给出内直径与外半径估计，进而在一般限制条件下刻画其拓扑性质。
Let (M,g) be a complete 3-dimensional asymptotically flat manifold with everywhere positive scalar curvature. We prove that, given a compact subset K of M, all volume preserving stable constant mean curvature surfaces of sufficiently large area will avoid K. This complements the work of G. Huisken and S.-T. Yau and J. Qing and G. Tian whose results show that large spherical volume preserving stable constant mean curvature spheres that omit a large compact subset of an asymptotically flat manifold that is $C^4$-asymptotic to positive mass Schwarzschild are uniquely determined by their area. The…
This paper concerns the evolution of a closed hypersurface of dimension $n(\geq 2)$ in the Euclidean space ${\mathbb{R}}^{n+1}$ under a mixed volume preserving flow. The speed equals a power $β(\geq 1)$ of homogeneous, either convex or concave, curvature functions of degree one plus a mixed volume preserving term, including the case of powers of the mean curvature and of the Gauss curvature. The main result is that if the initial hypersurface satisfies a suitable pinching condition, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces converge expone…
Given a compact manifold $M$ and a Riemannian manifold $N$ of bounded geometry, we consider the manifold ${\rm Imm} (M,N)$ of immersions from $M$ to $N$ and its subset ${\rm Imm}_μ(M,N)$ of those immersions with the property that the volume-form of the pull-back metric equals $μ$. We first show that the non-minimal elements of ${\rm Imm}_μ(M,N) $ form a splitting submanifold. On this submanifold we consider the Levi-Civita connection for various natural Sobolev metrics write down the geodesic equation and show local well-posedness in many cases. The question is a natural generalization of the …
This paper concerns closed hypersurfaces of dimension $n(\geq 2)$ in the hyperbolic space ${\mathbb{H}}_κ^{n+1}$ of constant sectional curvature $κ$ evolving in direction of its normal vector, where the speed is given by a power $β(\geq 1/m)$ of the $m$th mean curvature plus a volume preserving term, including the case of powers of the mean curvature and of the $\mbox{Gauß}$ curvature. The main result is that if the initial hypersurface satisfies that the ratio of the biggest and smallest principal curvature is close enough to 1 everywhere, depending only on $n$, $m$, $β$ and $κ$, then under t…
In this paper, we establish the rigidity result for local holomorphic volume preserving maps from an irreducible Hermitian manifold of compact type into its Cartesian products.
In this note, we derive a stability and weak-strong uniqueness principle for volume-preserving mean curvature flow. The proof is based on a new notion of volume-preserving gradient flow calibrations, which is a natural extension of the concept in the case without volume preservation recently introduced by Fischer et al. [arXiv:2003.05478]. The first main result shows that any strong solution with certain regularity is calibrated. The second main result consists of a stability estimate in terms of a relative entropy, which is valid in the class of distributional solutions to volume-preserving m…
We study the phase field method for the volume preserving mean curvature flow. Given an initial $C^1$ hypersurface we proved the existence of the weak solution for the volume preserving mean curvature flow via the reaction diffusion equation with a nonlocal term. We also show the monotonicity formula and the density upper bound for the reaction diffusion equation.
We prove that any real-analytic, volume-preserving action of a lattice $Γ$ in a simple Lie group with $\Qrank(Γ)\geq 7$ on a closed 4-manifold of nonzero Euler characteristic factors through a finite group action.
We study stable surfaces, i.e., second order minima of the area for variations of fixed volume, in sub-Riemannian space forms of dimension $3$. We prove a stability inequality and provide sufficient conditions ensuring instability of volume-preserving area-stationary $C^2$ surfaces with a non-empty singular set of curves. Combined with previous results, this allows to describe any complete, orientable, embedded and stable $C^2$ surface $Σ$ in the Heisenberg group $\mathbb{H}^1$ and the sub-Riemannian sphere $\mathbb{S}^3$ of constant curvature $1$. In $\mathbb{H}^1$ we conclude that $Σ$ is a E…
We consider the evolution of a closed convex hypersurface under a volume preserving curvature flow. The speed is given by a power of the m-th mean curvature plus a volume preserving term, including the case of powers of the mean curvature or of the Gauss curvature. We prove that if the initial hypersurface satisfies a suitable pinching condition, the solution exists for all times and converges to a round sphere.
Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space
本文研究双曲空间中凸超曲面的保体积高斯曲率流，证明解长期存在且指数收敛到测地球，是首个仅要求初始凸性的非局部型保体积曲率流结果。
Curvature measures and volume preserving flows in h-convex geometry
本文针对双曲空间h-凸几何，引入修正曲率测度概念并推广经典结果，研究保体积移位α-高斯曲率流，证明其解全局存在且收敛到测地球，修正曲率测度理论是刻画极限形状的关键。
We show that every volume preserving codimension one Anosov flow on a closed Riemannian manifold of dimension greater than three admits a global cross section and is therefore topologically conjugate to a suspension of a linear toral automorphism. This proves a conjecture of Verjovsky from the 1970's in the volume preserving case.
In this paper,we obtain two results on closed Reimainnian manifold $M\times [0,T]$.When $T$ is small enough,to any prescribed scalar curvature, the existence and uniqueness of metrics are obtained on the volume element preserving deformation.When $T$ is large and the given scalar curvature is small enough,the same result holds.
For a germ of a smooth map f and a subgroup G_V of any of the Mather groups G for which the source or target diffeomorphisms preserve some given volume form V in the source or in the target we study the G_V-moduli space of f that parameterizes the G_V-orbits inside the G-orbit of f. We find, for example, that this moduli space vanishes for A-equivalence with volume-preserving target diffeomorphisms and A-stable maps f and for K-equivalence with volume-preserving source diffeomorphisms and K-simple maps f. On the other hand, there are A-stable maps f with infinite-dimensional moduli space for A…
We extend V. Arnold's theory of asymptotic linking for two volume preserving flows on a domain in ${\mathbb R}^3$ and $S^3$ to volume preserving actions of ${\mathbb R}^k$ and ${\mathbb R}^\ell$ on certain domains in ${\mathbb R}^n$ and also to linking of a volume preserving action of ${\mathbb R}^k$ with a closed oriented singular $\ell$-dimensional submanifold in ${\mathbb R}^n$, where $n=k+\ell+1$. We also extend the Biot-Savart formula to higher dimensions.
We consider curvature flows in hyperbolic space with a monotone, symmetric, homogeneous of degree 1 curvature function F. Furthermore we assume F to be either concave and inverse concave or convex. For compact initial hypersurfaces, which are strictly convex by horospheres, we show the long time existence of mixed volume preserving curvature flows. Furthermore we show that these flows converge exponentially in the C-infinity topology to a geodesic sphere.
We consider a closed surface in $\mathbb{R}^3$ evolving by the volume-preserving Willmore flow and prove a lower bound for the existence time of smooth solutions. For spherical initial surfaces with Willmore energy below $8π$ we show long time existence and convergence to a round sphere by performing a suitable blow-up and by proving a constrained Lojasiewicz-Simon inequality.
A sharp quantitative Alexandrov inequality and applications to volume preserving geometric flows in 3D
本文研究三维保体积平均曲率流和 Mullins-Sekerka 流的渐近行为，并建立了带周长界的 $C^2$ 集合的尖锐定量 Alexandrov 不等式，为其应用提供理论基础。
A dual pair for the group of volume preserving diffeomorphisms
该研究利用光滑嵌入空间的余切丛构造保体积微分同胚群的辛对偶对，通过辛约化用增广子流形的非线性格拉斯曼流形描述其馀伴随轨道，并给出余维数1嵌入下子流形的约束条件。
We consider the dynamic property of the volume preserving mean curvature flow. This flow was introduced by Huisken who also proved it converges to a round sphere of the same enclosed volume if the initial hypersurface is strictly convex in Euclidean space. We study the stability of this flow in hyperbolic space. In particular, we prove that if the initial hypersurface is hyperbolically mean convex and close to an umbilical sphere in the $L^2$-sense, then the flow exists for all time and converges exponentially to an umbilical sphere.
Asymptotic behavior of a diffused interface volume-preserving mean curvature flow
本文研究了欧氏空间中保体积平均曲率流的扩散界面版本，证明了在所有维度及自然初始假设下，该流指数收敛至单个“扩散球”。
We investigate the formation of singularities for surfaces evolving by volume preserving mean curvature flow. For axially symmetric flows - surfaces of revolution - in $\mathbb{R}^3$ with Neumann boundary conditions, we prove that the first developing singularity is of Type I. The result is obtained without any additional curvature assumptions being imposed, while axial symmetry and boundary conditions are justifiable given the volume constraint. Additional results and ingredients towards the main proof include a non-cylindrical parabolic maximum principle, and a series of estimates on geometr…
We provide a variational description of any Liouville (i.e. volume preserving) autonomous vector fields on a smooth manifold. This is obtained via a ``maximal degree'' variational principle; critical sections for this are integral manifolds for the Liouville vector field. We work in coordinates and provide explicit formulae.
We study the exponential map of group of volume-preserving diffeomorphisms on closed orientable surfaces via the vorticity formulation of the incompressible Euler equation. We present an alternative, fluid dynamical proof of the theorem of Ebin--Misiołek--Preston: the exponential is a nonlinear Fredholm mapping of index zero. We extend Shnirelman's rigidity result for the exponential map from 2-dimensional flat torus to arbitrary orientable closed surfaces. That is, we prove that the exponential map is Fredholm quasiregular.
Decay of correlations on Abelian covers of isometric extensions of volume-preserving Anosov flows
研究了封闭流形阿贝尔覆盖上保体积阿诺索夫流的各向同性扩展的关联函数的渐近衰减行为。
I construct the real counterparts (which I call Borel-Bott classes) of the R/Z classes constructed in "Characteristic classes in symplectic topology", to appear, in the cohomology of volume-preserving and symplectomorhisms of a compact (symplectic) manifold.I show that, for the symplectic action of the mapping class group in the moduli space of stable vector bundles over a Riemann surface, the restriction of the first constructed class from the symplectomorphism group gives a generator for the second (bounded) cohomology of the mapping class group.
In this article, we continue the work in \cite{GL} and study a normalized hypersurface flow in the more general ambient setting of warped product spaces. This flow preserves the volume of the bounded domain enclosed by a graphical hypersurface, and monotonically decreases the hypersurface area. As an application, the isoperimetric problem in warped product spaces is solved for such domains.
We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional $\mathcal I$ defined on exact divergence-free vector fields of class $C^1$ on a compact 3-manifold that is associated with a well-behaved integral kernel, we prove that $\mathcal I$ is invariant under arbitrary volume-preserving diffeomorphisms if and only if it is a function of the helicity.
Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow
该研究提出半空间中带毛细边界超曲面的保体积（或保面积）幂平均曲率流，并证明任意凸初始超曲面均可全局存在且光滑收敛至球冠。
We show that the discrete approximate volume preserving mean curvature flow in the flat torus $\mathbb{T}^N$ starting near a strictly stable critical set $E$ of the perimeter converges in the long time to a translate of $E$ exponentially fast. As an intermediate result we establish a new quantitative estimate of Alexandrov type for periodic strictly stable constant mean curvature hypersurfaces. Finally, in the two dimensional case a complete characterization of the long time behaviour of the discrete flow with arbitrary initial sets of finite perimeter is provided.
We consider the flow of closed convex hypersurfaces in Euclidean space $\mathbb{R}^{n+1}$ with speed given by a power of the $k$-th mean curvature $E_k$ plus a global term chosen to impose a constraint involving the enclosed volume $V_{n+1}$ and the mixed volume $V_{n+1-k}$ of the evolving hypersurface. We prove that if the initial hypersurface is strictly convex, then the solution of the flow exists for all time and converges to a round sphere smoothly. No curvature pinching assumption is required on the initial hypersurface.
In this paper we consider a class of weighted-volume preserving curvature flows acting on hypersurfaces that are trapped within two parallel hyperplanes and satisfy an orthogonal boundary condition. In the author's thesis the stability of cylinders under the flows was considered; it was found that they are stable provided the radius satisfies a certain condition. Here we consider flows where there is a critical radius such that the cylinders are only stable on one side of the critical value, and the weight function is a linear combination of the elementary symmetric functions, the reason for s…
Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form. We show that every homogeneous quasi-morphism on the identity component $Diff_0(M,vol)$ of the group of volume preserving diffeomorphisms of M, which is induced by a quasi-morphism on the fundamental group, is Lipschitz with respect to the L^p-metric on the group $Diff_0(M,vol)$. As a consequence, assuming certain conditions on the fundamental group, we construct bi-Lipschitz embeddings of finite dimensional vector spaces into $Diff_0(M,vol)$.
Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness
本文提出了一种新的 varifold 解概念，用于双相体积保持均 curvature 流，证明了其存在性和弱强唯一性。
A smooth isotopy of volume-preserving diffeomorphisms on unit cube saving energy through extra dimensions
该研究提出了一种在单位立方体上通过额外维度实现能量保存的体积保持微分同胚同伦，并展示了其与有限动能同伦的区别。
Center manifold analysis can be used in order to investigate the stability of the stationary solutions of various PDEs. This can be done by considering the PDE as an ODE between certain Banach spaces and linearising about the stationary solution. Here we investigate the volume preserving mean curvature flow using such a technique. We will consider surfaces with boundary contained within two parallel planes such that the surface meets these planes orthogonally. With this set up the stationary solution is a cylinder. We will find that for initial surfaces that are sufficiently close to a cylinde…
Length distortion of volume-preserving Lipschitz mappings
本文证明度量曲面间保面积Lipschitz映射几乎不畸变曲线长度，给出两种证明方法，并推广到高维及非光滑情形。
On dynamical invariants of coadjoint orbits of 3D volume-preserving diffeomorphisms
本文研究三维保体积向量场中螺旋度唯一性的失效程度，证明在整同调球面的 $C^1$ 拓扑下存在大量独立于螺旋度的连续全局动力学不变量，从而否定回答了 Arnold 和 Khesin 关于余伴随轨道稠密性的问题。
Volume preserving spacetime mean curvature flow and foliations of initial data sets
本文研究具有正ADM能量的渐近欧几里得初始数据集中体积保持时空平均曲率流，证明其长期存在并收敛到常数曲率极限，为CSTMC叶状结构提供新构造，应用于广义相对论中孤立系统质心的定义。
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