We show the uniqueness of strictly convex closed smooth self-similar solutions to the $α$-Gauss curvature flow with $(1/n) < α< 1+(1/n)$. We introduce a Pogorelov type computation, and then we apply the strong maximum principle. Our work combined with earlier works on the Gauss Curvature flow imply that the $α$-Gauss curvature flow with $(1/n) < α< 1+(1/n)$ shrinks a strictly convex closed smooth hypersurface to a round sphere.
In this paper, we discuss uniqueness and backward uniqueness for mean curvature flow of non-compact manifolds. We use an energy argument to prove two uniqueness theorems for mean curvature flow with possibly unbounded curvatures. These generalize the results by Chen and Yin. Using similar method, we also obtain a uniqueness result on Ricci flows. A backward uniqueness theorem is also proved for mean curvature flow with bounded curvatures.
In this paper, we prove that the transverse Mabuchi K-energy functional is convex along the weak geodesic in the space of Sasakian metrics. As an application, we obtain the uniqueness of constant scalar curvature Sasakian metrics modulo automorphisms for the transverse holomorphic structure.
The aim of this paper is to give two uniqueness results for the Dirichlet problem associated to the constant mean curvature equation. We study constant mean curvature graphs over strips of R^2. The proofs are based on height estimates and the study of the asymptotic behaviour of solutions to the Dirichlet problem.
The Skew Mean Curvature Flow(SMCF) is a Schrödinger-type geometric flow canonically defined on a co-dimension two submanifold, which generalizes the famous vortex filament equation in fluid dynamics. In this paper, we prove the local existence and uniqueness of general dimensional SMCF in Euclidean spaces.
In this paper, we consider a family of closed hypersurfaces which shrink self-similarly with speed of quotient curvatures. We show that the only such hypersurfaces are shrinking spheres.
In this paper I study the constant mean curvature surface in asymptotically flat 3-manifolds with general asymptotics. Under some weak condition, I prove that outside some compact set in the asymptotically flat 3-manifold with positive mass, the foliation of stable spheres of constant mean curvature is unique.
The recently introduced Lipschitz-Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz-Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.
In this paper we show that an immersed nontrivial translating soliton for mean curvature flow in $\mathbb{R}^{n+1}$($n=2,3)$ is a grim hyperplane if and only if it is mean convex and has weighted total extrinsic curvature of at most quadratic growth. For an embedded translating soliton $Σ$ with nonnegative scalar curvature, we prove that if the mean curvature of $Σ$ does not change signs on each end, then $Σ$ must have positive scalar curvature unless it is either a hyperplane or a grim hyperplane.
In this note, we study the problem of uniqueness of Ricci flow on complete noncompact manifolds. We consider the class of solutions with curvature bounded above by C/t when t > 0. In paricular, we proved uniqueness if in addition the initial curvature is of polynomial growth and Ricci curvature of the flow is relatively small.
In this paper we study the uniqueness of graphical mean curvature flow. We consider as initial conditions graphs of locally Lipschitz functions and prove that in the one dimensional case solutions are unique without any further assumptions. This result is then generalized for rotationally symmetric solutions. In the general $n$- dimensional case, we prove uniqueness under additional conditions: we require a { \em uniform lower bound } on the second fundamental form and the height function of the initial condition. The latter result extends to initial conditions that are proper graphs over subd…
In this paper we study the mean curvature flow of embedded disks with free boundary on an embedded cylinder or generalised cone of revolution, called the support hypersurface. We determine regions of the interior of the support hypersurface such that initial data is driven to a curvature singularity in finite time or exists for all time and converges to a minimal disk. We further classify the type of the singularity. We additionally present applications of these results to the uniqueness problem for minimal hypersurfaces with free boundary on such suppport hypersurfaces; the results obtained t…
In this paper, we can prove the existence and uniqueness of solutions to the constant mean curvature (CMC for short) equation with nonzero Neumann boundary data in product manifold $M^{n}\times\mathbb{R}$, where $M^{n}$ is an $n$-dimensional ($n\geq2$) complete Riemannian manifold with nonnegative Ricci curvature, and $\mathbb{R}$ is the Euclidean $1$-space. Equivalently, this conclusion gives the existence of CMC graphic hypersurfaces defined over a compact strictly convex domain $Ω\subset M^{n}$ and having arbitrary contact angle.
In this paper we consider closed non-collapsed ancient solutions to the mean curvature flow ($n \ge 2$) which are uniformly two-convex. We prove that any two such ancient solutions are the same up to translations and scaling. In particular, they must coincide up to translations and scaling with the rotationally symmetric closed ancient non-collapsed solution constructed by Brian White in (2000), and by Robert Haslhofer and Or Hershkovits in (2016).
We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed spheres of prescribed Weingarten curvature in R3 for the special but important case of prescribed mean curvature. As a consequence, we extend the classical Hopf uniqueness theorem for constant mean curvature spheres to the case of immersed spheres of prescribed antipodally symmetric mean curvature in R3.
We show that every convex ancient solution of mean curvature flow with Type I curvature growth is either spherical, cylindrical, or planar. We then prove the corresponding statement for flows by a natural class of curvature functions which are convex or concave in the second fundamental form. Neither of these results assumes interior noncollapsing.
The aim of this paper is to investigate uniqueness of conic constant scalar curvature Kaehler (cscK) metrics, when the cone angle is less than $π$. We introduce a new Hölder space called $\cC^{4,\a,\b}$ to study the regularities of this fourth order elliptic equation, and prove that any $\cC^{2,\a,\b}$ conic cscK metric is indeed of class $\cC^{4,\a,\b}$. Finally, the reductivity is established by a careful study of the conic Lichnerowicz operator.
In Riemannian geometry the prescribed Ricci curvature problem is as follows: given a smooth manifold $M$ and a symmetric 2-tensor $r$, construct a metric on $M$ whose Ricci tensor equals $r$. In particular, DeTurck and Koiso proved the following celebrated result: the Ricci curvature uniquely determines the Levi-Civita connection on any compact Einstein manifold with non-negative section curvature. In the present paper we generalize the result of DeTurck and Koiso for a Riemannian manifold with non-negative section curvature. In addition, we extended our result to complete non-compact Riemanni…
We demonstrate that the uniqueness of solutions to a broad class of parabolic geometric evolution equations can be proven via a direct and essentially classical energy argument which avoids the DeTurck trick entirely. Previously, we have used a variation of this technique to give an alternative proof and slight extension to the basic uniqueness result for complete solutions to the Ricci flow of uniformly bounded curvature. Here we extend this approach to curvature flows of all orders, including the $L^2$-curvature flow and a class of quasilinear higher-order flows related to the obstruction te…
It has been known in that round spheres are the only closed homothetic self-similar solutions to the inverse mean curvature flow and parabolic curvature flows by degree -1 homogeneous functions of principle curvatures in the Euclidean space. In this article, we prove that the round sphere is rigid in much stronger sense, that under some natural conditions such as star-shapedness, it is the only closed solution to the inverse mean curavture flow and the above-mentioned flows in the Euclidean space which evolves by diffeomorphisms generated by conformal Killing fields.
We study the classification of immersed constant mean curvature (CMC) spheres in the homogeneous Riemannian 3-manifold Sol_3, i.e., the only Thurston 3-dimensional geometry where this problem remains open. Our main result states that, for every H>1/(\sqrt{3}), there exists a unique (up to left translations) immersed CMC H sphere S_H in Sol_3 (Hopf-type theorem). Moreover, this sphere S_H is embedded, and is therefore the unique (up to left translations) compact embedded CMC H surface in Sol_3 (Alexandrov-type theorem). The uniqueness parts of these results are also obtained for all real number…
We study uniqueness of positive solutions to the conformal scalar curvature equation on complete Riemannian manifolds with constant negative scalar curvature. We apply the results to show that conformal transformations on certain complete Riemannian manifolds of constant negative scalar curvature are isometries. We also study uniqueness of complete positive solutions and radial solutions.
In this paper, we give a height estimate for constant mean curvature graphs. Using this result we prove two results of uniqueness for the Dirichlet problem associated to the constant mean curvature equation on unbounded domains.
In this paper, we prove the uniqueness of asymptotically conical tangent flows in all codimensions. This is based on an early work of Chodosh-Schulze, who proved the uniqueness in the hypersurface case.
Mean curvature flow evolves isometrically immersed base manifolds $M$ in the direction of their mean curvatures in an ambient manifold $\bar{M}$. If the base manifold $M$ is compact, the short time existence and uniqueness of the mean curvature flow are well-known. For complete isometrically immersed submanifolds of arbitrary codimensions, the existence and uniqueness are still unsettled even in the Euclidean space. In this paper, we solve the uniqueness problem affirmatively for the mean curvature flow of general codimensions and general ambient manifolds. In the second part of the paper, ins…
We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least 5. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which the transversal Futaki invariant (restricted to the Lie algebra of the torus) vanishes. Using existence result of [25], we show that a co-oriented compact toric contact 5-manifold whose moment cone has 4 facets admits a finite number of rays of transversal homothetic compatible …
We show existence of ancient solutions to the rescaled mean curvature flow starting from a given asymptotically conical self-expander. These are examples of mean curvature flows coming out of cones that are not self-similar. We also show a strong uniqueness theorem when the cone is generic and use it to classify mean curvature flows coming out of generic cones of small entropy in low dimensions.
We show short time existence and uniqueness of $\C^{1,1}$ solutions to the mean curvature flow with obstacles, when the obstacles are of class $\C^{1,1}$. If the initial interface is a periodic graph we show long time existence of the evolution and convergence to a minimal constrained hypersurface.
We revisit the problem of uniqueness for the Ricci flow and give a short, direct proof, based on the consideration of a simple energy quantity, of Hamilton/Chen-Zhu's theorem on the uniqueness of complete solutions of uniformly bounded curvature. With a variation of this quantity and technique, we further prove a uniqueness theorem for subsolutions to a general class of mixed differential inequalities which implies an extension of Chen-Zhu's result to solutions (and initial data) of potentially unbounded curvature.
Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, has at most one solution in the class of complete riemannian metric with complex sectional curvature bounded from below.
We prove that a H-surface M in H^2xR, |H| <= 1/2, inherits the symmetries of its boundary when the boundary is either a horizontal curve with curvature greater than one or two parallel horizontal curves with curvature greater than one, whose distance is greater or equal to πFurthermore we prove that the asymptotic boundary of a surface with mean curvature bounded away from zero consists of parts of straight lines, provided it is sufficiently regular
We give a simple, direct proof of the backward uniqueness of solutions to a class of second-order geometric evolution equations including the Ricci and cross-curvature flows. The proof, based on a classical argument of Agmon-Nirenberg, uses the logarithmic convexity of a certain energy quantity in the place of Carleman inequalities. We also demonstrate the applicability of the technique to the $L^2$-curvature flow and other higher-order equations.
An anisotropic surface energy is the integral of an energy density that depends on the normal at each point over the considered surface, and it is a generalization of surface area. The minimizer of such an energy among all closed surfaces enclosing the same volume is unique and it is (up to rescaling) so-called the Wulff shape. We prove that, unlike the isotropic case, there exists an anisotropic energy density function such that there exist closed embedded equilibrium surfaces with genus zero in the three-dimensional Euclidean space each of which is not (any homothety and translation of) the …
We prove that constant scalar curvature Kähler metric "adjacent" to a fixed Kähler class is unique up to isomorphism. This extends the uniqueness theorem of Donaldson and Chen-Tian, and formally fits into the infinite dimensional G.I.T picture described by Donaldson. We prove that the Calabi flow near a cscK metric exists globally and converges uniformly to a cscK metric in a polynomial rate. Viewed in a Kähler class, the Calabi flow is also shown to be asymptotic to a smooth geodesic ray at infinity. This latter fact is also interesting in the finite dimensional analogue, where we show that t…
In this note we prove uniqueness of nondegenerate compact blowups for the motion by curvature of planar networks. The proof follows ideas introduced in "Lojasiewicz-Simon inequalities for minimal networks: stability and convergence" for the study of stability properties of critical points of the length functional and it is based on the application of a Lojasiewicz-Simon gradient inequality.
In this note we prove the backwards uniqueness of the mean curvature flow for (codimension one) hypersurfaces in a Euclidean space. More precisely, let $F_t, \widetilde{F}_t:M^n \rightarrow \mathbb{R}^{n+1}$ be two complete solutions of the mean curvature flow on $M^n \times [0,T]$ with bounded second fundamental forms. Suppose $F_T=\widetilde{F}_T$, then $F_t=\widetilde{F}_t$ on $M^n \times [0,T]$. This is an analog of a result of Kotschwar on the Ricci flow.
We show that a minimal disk satisfying the free boundary condition in a constant curvature ball of any dimension is totally geodesic. We weaken the condition to parallel mean curvature vector in which case we show that the disk lies in a three dimensional constant curvature submanifold and is totally umbilic. These results extend to higher dimensions earlier three dimensional work of J. C. C. Nitsche and R. Souam.
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 $κ$-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.
We classify static manifolds which admit more than one static decomposition whenever a condition on the curvature is fullfilled. For this, we take a standard static vector field and analyze its associated one parameter family of projections onto the base. We show that the base itself is a static manifold and the warping function satisfies severe restrictions, leading us to our classification results. Moreover, we show that certain condition on the lightlike sectional curvature ensures the uniqueness of static decomposition for Lorentzian manifolds.
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…
Let $ψ$ be a given function defined on a Riemannian space. Under what conditions does there exist a compact starshaped hypersurface $M$ for which $ψ$, when evaluated on $M$, coincides with the $m-$th elementary symmetric function of principal curvatures of $M$ for a given $m$? The corresponding existence and uniqueness problems in Euclidean space have been investigated by several authors in the mid 1980's. Recently, conditions for existence were established in elliptic space and, most recently, for hyperbolic space. However, the uniqueness problem has remained open. In this paper we investigat…
We state that any constant curvature Riemannian metric with conical singularities of constant sign curvature on a compact (orientable) surface $S$ can be realized as a convex polyhedron in a Riemannian or Lorentzian) space-form. Moreover such a polyhedron is unique, up to global isometries, among convex polyhedra invariant under isometries acting on a totally umbilical surface. This general statement falls apart into 10 different cases. The cases when $S$ is the sphere are classical.
We study a variational problem for piecewise-smooth hypersurfaces in the (n+1)-dimensional Euclidean space with an anisotropic energy. An anisotropic energy is the integral of an energy density that depends on the normal at each point over the considered hypersurface. The minimizer of such an energy among all closed hypersurfaces enclosing the same (n+1)-dimensional volume is unique and it is (up to rescaling) so-called the Wulff shape. The Wulff shape and equilibrium hypersurfaces of this energy for volume-preserving variations are not smooth in general. We prove that, if the anisotropic ener…
We prove existence and uniqueness of foliations by stable spheres with constant mean curvature for 3-manifolds which are asymptotic to Anti-de Sitter-Schwarzschild metrics with positive mass. These metrics arise naturally as spacelike timeslices for solutions of the Einstein equation with a negative cosmological constant.
In a previous paper, the authors showed that metrics which are asymptotic to Anti-de Sitter-Schwarzschild metrics with positive mass admit a unique foliation by stable spheres with constant mean curvature. In this paper we extend that result to all asymptotically hyperbolic metrics for which the trace of the mass term is positive. We do this by combining the Kazdan-Warner obstructions with a theorem due to De Lellis and Müller.
