Stability of capillary hypersurfaces with higher order mean curvature.
problem Stability of capillary hypersurfaces with constant higher order mean curvature.
method Generalization of classical stability theory for capillary hypersurfaces.
result Results on stability for capillary hypersurfaces with higher order mean curvature.
Study higher order mean curvatures in SAC half-lightlike submanifolds.
problem Exploring new geometric properties of SAC submanifolds.
method Introduced and used higher order mean curvatures; derived integration formulae.
result Generalized known results and derived new integration formulae.
Study on uniqueness of hypersurfaces in hyperbolic space with constant mean curvature.
problem Uniqueness of hypersurfaces with constant higher order mean curvature in hyperbolic space.
method Generalization of Bernstein theorem and proof of Bernstein type results for immersed hypersurfaces.
result Rigidity of horospheres and equidistant spheres in terms of their higher order mean curvatures.
We obtain sharp estimates involving the mean curvatures of higher order of a complete bounded hypersurface immersed in a complete Riemannian manifold. Similar results are also given for complete spacelike hypersurfaces in Lorentzian ambient spaces.
In this article, using the generalized Newton transformation, we define higher order mean curvatures of distributions of arbitrary codimension and we show that they agree with the ones from Brito and Naveira (Ann. Global Anal. Geom. 18, 371-383 (2000)). We also introduce higher order mean curvature vector fields and we…
In this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker (GRW) spacetimes. In particular, we consider the following question: Under what conditions must a compact spacelike hypersurface with constant higher order mean curvatur…
The paper proves existence and classification of translating solitons in warped product manifolds.
problem Existence and classification of translating solitons in warped product manifolds.
method Proving existence and classification results for translating solitons defined as initial conditions for higher order mean curvature flows in warped product manifolds.
result Existence and classification of translating solitons in warped product manifolds.
The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codim…
In this paper we characterize compact and complete hypersurfaces with some constant higher order mean curvature into warped product spaces. Our approach is based on the use of a new trace operator version of the Omori-Yau maximum principle which seems to be interesting in its own.
Classifies hypersurfaces with positive constant mean curvature in hyperbolic space.
problem Classifying hypersurfaces with positive constant mean curvature in hyperbolic space.
method Classifies hypersurfaces with rotational symmetry and positive constant r-th mean curvature in HnimesR. result Compact connected hypersurfaces of constant r-th mean curvature embedded in Hnimes[0,∞) with boundary in the slice Hnimes{0} are topological disks under suitable assumptions. Motivated by the theory of isoparametric hypersurfaces, we study submanifolds whose tubular hypersurfaces have some constant "higher order mean curvatures". Here a k-th order mean curvature Qk (k≥1) of a hypersurface Mn is defined as the k-th power sum of the principal curvatures, or equivalently, of the…
Study soap bubbles with almost constant higher-order mean curvature, proving unique limits under certain conditions.
problem Understanding the asymptotic behavior of soap bubbles with almost constant higher-order mean curvature.
method Analyzing sequences of bounded C2-domains in Rn+1 converging in volume and perimeter, with k-th mean curvature functions converging in L1. result Finite unions of mutually tangent balls are the only possible limits under natural mean convexity and L∞-control on the mean curvature outside a set of vanishing area. The paper extends sphere theorems to higher-order mean curvature functions on specific hypersurfaces.
problem Proving sphere theorems for hypersurfaces with W2,n regularity. method Extending Montiel-Ros argument and using Legendrian cycles.
result Proves existence of n-dimensional Legendrian cycles with 2n-dimensional support. New findings on hypersurfaces with specific curvature properties in space forms.
problem Characterizing hypersurfaces with almost constant curvature in space forms.
method Analyzing starshaped hypersurfaces with various curvature conditions.
result Closed starshaped hypersurfaces with almost constant mean curvature or higher order mean curvature are close to geodesic spheres.
The paper proves smoothness of mean curvature flow with triple junctions.
problem Mean curvature flow with triple junctions and higher order junctions.
method Extending classical results to flows with triple junctions, showing smoothness for weakly close flows.
result Smooth short-time existence of mean curvature flow with triple junctions.
Based on ideas of L. Alías, D. Impera and M. Rigoli developed in "Hypersurfaces of constant higher order mean curvature in warped products", we develope a fairly general weak/Omori-Yau maximum principle for trace operators. We apply this version of maximum principle to generalize several higher order mean curvature est…
We introduce a regularization method for mean curvature flow of a submanifold of arbitrary codimension in the Euclidean space, through higher order equations. We prove that the regularized problems converge to the mean curvature flow for all times before the first singularity.
The Clifford torus is unstable but rigid in mean curvature flow.
problem Stability and rigidity of the Clifford torus in mean curvature flow.
method Analysis of higher order phenomena, including entropy minimisation and infinitesimal deformations.
result The Clifford torus is locally unique as a self-shrinker for mean curvature flow.
In this paper we analyze the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes. We consider first the case of compact spacelike hypersurfaces, completing some previous results given in [2]. We next extend these results to the complete …
The paper proves a new discrete Laplacian for 3D meshes and shows its superiority over primal construction.
problem Developing a more accurate discrete Laplacian for 3D meshes.
method Proves the Euler-Lagrange equation for the Dirichlet energy using the associated discrete Laplacian of the dual construction.
result The associated discrete Laplacian is optimal in R3 compared to the primal construction. We find complete hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of (elliptic) curvature functions which includes the higher order mean curvatures and their curvature quotients.
It is still an open question whether a compact embedded hypersurface in the Euclidean space R^{n+1} with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or a spherical cap, even in the simplest case of surfaces in R^3. In a recent paper the first and third authors have shown that this i…
The study finds translators for higher order mean curvature flows in Euclidean and hyperbolic spaces.
problem Finding translators for higher order mean curvature flows in different spaces.
method Analyzing velocity functions of translators to r-mean curvature flows in RnimesR and HnimesR. result Existence and uniqueness of translators, including bowl-type, catenoid-type, and Grim Reaper-type translators.
In this paper we obtain a sharp height estimate concerning compact hypersurfaces immersed into warped product spaces with some constant higher order mean curvature, and whose boundary is contained into a slice. We apply these results to draw topological conclusions at the end of the paper.
In this paper, we first investigate several rigidity problems for hypersurfaces in the warped product manifolds with constant linear combinations of higher order mean curvatures as well as "weighted'' mean curvatures, which extend the work \cite{Mon, Brendle,BE} considering constant mean curvature functions. Secondly, …
Study shows constant curvature convex hypersurfaces on hyperboloids are parts of hyperboloids.
problem Characterizing convex hypersurfaces with constant curvature on hyperboloids.
method Analyzing hypersurfaces with constant higher order mean curvature and constant boundary angle.
result Hypersurfaces with constant curvature on hyperboloids are parts of hyperboloids.
New proof shows symmetry for certain curved surfaces in higher dimensions.
problem Understanding symmetries in curved surfaces evolving over time.
method Generalized previous result to higher dimensions, proving symmetry condition.
result Uniformly 3-convex translating solitons must have SO(n−1) symmetry. Proves higher regularity for anisotropic inverse mean curvature flow.
problem Higher regularity of solutions to anisotropic inverse mean curvature flow.
method Proves Harnack estimate and constructs smooth solutions from C1 initial sets. result Smooth solutions become smooth outside a compact set.
Maximal mean curvature limits surface area in higher dimensions.
problem Maximal mean curvature limits surface area in higher dimensions.
method Smooth embeddings of the ball with arbitrary small volume for given maximal mean curvature.
result For a given maximal mean curvature, smooth embeddings of the ball with arbitrary small volume are provided.
Constructs conformal boundary operators and fractional Laplacians.
problem Developing conformally invariant boundary operators and fractional Laplacians.
method Constructs continuously parametrised families of conformally invariant boundary operators on densities.
result Constructs odd-order conformally invariant fractional Laplacian pseudo-differential operators.
Global bounds for mean curvature flow gradients are derived.
problem Global regularity of mean curvature flows in higher dimensions.
method Derives global bounds for Hölder norms of gradients of solutions.
result Global bounds for Hölder norms of gradients are derived.
Expands differential geometry to higher-order infinitesimals.
problem No specific problem stated; general expansion of differential geometry.
method Introduces higher tangent vectors and jet connections, generalizes Riemannian metric tensor, develops higher-order integration theory.
result Natural analogues of Riemannian curvature tensor with novel phenomena.
In this paper we prove Hessian and Laplacian comparison theorems for the Lorentzian distance function in a spacetime with sectional (or Ricci) curvature bounded by a certain function by means of a comparison criterion for Riccati equations. Using these results, under suitable conditions, we are able to obtain some esti…
Study proves rigidity theorems for ancient solutions to mean curvature flow with convex image.
problem Rigidity of ancient solutions to mean curvature flow with convex Gauss image.
method Refined curvature estimates.
result Better rigidity theorems for ancient solutions in higher codimension.
The paper proves unique ancient solutions to mean curvature flow in higher dimensions are symmetric.
problem Proving uniqueness of ancient solutions to mean curvature flow in higher dimensions.
method Analyzing strictly convex, uniformly two-convex, and noncollapsed ancient solutions.
result Ancient solutions are rotationally symmetric translating solitons.
Given a regular bounded domain Ω⊂R2m, we describe the limiting behavior of sequences of solutions to the mean field equation of order 2m, m≥1, (−Δ)mu=ρ∫Ωe2mudxe2muinΩ, under the Dirichlet boundary condition and the bound 0<ρ≤C. We emphasize the connection wi…
The paper provides estimates for higher-order Ricci curvature along Kähler-Ricci flows.
problem Estimating higher-order curvature along Kähler-Ricci flows on compact Kähler manifolds.
method Proving uniform bounds for Ricci curvature and scalar curvature in various orders and norms.
result A geometric obstruction causes a specific third-order derivative of Ricci curvature to blow up at rate et/2. The study proves a neighborhood theorem for mean curvature flow in higher dimensions.
problem Proving a canonical neighborhood theorem for mean curvature flow in higher dimensions.
method Proved a canonical neighborhood theorem for mean curvature flow of compact submanifolds in RN with a pinching condition. result Proved a canonical neighborhood theorem for mean curvature flow in dimensions n≥5. New characterization of geodesic spheres in space forms.
problem Characterize geodesic spheres in space forms.
method Utilizes the Omori-Yau maximum principle, Walter's formula for mean curvature, and Gårding's inequality.
result Geodesic spheres are the only complete bounded hypersurfaces with constant mean and scalar curvature.
Maps between 2D spaces evolve under area-decreasing conditions.
problem Understanding the evolution of maps under area-decreasing constraints.
method Mean curvature flow of the graph of a map between 2D Euclidean spaces.
result Existence and uniform decay estimates for the evolving submanifold.
Study proves mean curvature flows on spheres in higher dimensions.
problem Existence of mean curvature flows on spheres.
method Generalized previous results to higher dimensions, proving existence of flows.
result Existence of infinitely many eternal weak mean curvature flows in Sn+1 connecting specific hypersurfaces. Study of mean curvature flows on graphs in warped product manifolds, focusing on behavior at infinity.
problem Behavior of mean curvature flows on graphs in warped product manifolds, especially at infinity.
method Analysis of curve shortening flow and mean curvature flow on geodesic graphs for various warping functions.
result Long-time existence of mean curvature flows and vanishing of curvature and derivatives at infinity.
Given a positive function F on Sn which satisfies a convexity condition, for 1≤r≤n, we define the r-th anisotropic mean curvature function HrF for hypersurfaces in Rn+1 which is a generalization of the usual r-th mean curvature function. We prove that a compact embedded hypersurface…
The paper studies stable surfaces with constant curvature in 3D space forms.
problem Stability of surfaces with constant extrinsic curvature in space forms.
method Using stability notions for surfaces with constant higher order mean curvature.
result Rigidity results for surfaces with free boundary in geodesic balls or slabs.
We construct new examples of algebraic curvature tensors so that the Jordan normal form of the higher order Jacobi operator is constant on the Grassmannian of subspaces of type (r,s) in a vector space of signature (p,q). We then use these examples to establish some results concerning higher order Osserman and highe…
We derive pointwise curvature estimates for graphical mean curvature flows in higher codimensions. To the best of our knowledge, this is the first such estimates without assuming smallness of first derivatives of the defining map. An immediate application is a convergence theorem of the mean curvature flow of the graph…
The study finds inequalities in warped product manifolds linking isoperimetric and curvature integrals.
problem Finding inequalities in warped product manifolds.
method Proving isoperimetric inequalities and relating them to curvature integrals.
result Sharp eigenvalue and geometric inequalities in space forms.
The paper generalizes Alexandrov theorems for null hypersurfaces with integral curvature conditions.
problem Determining when a submanifold lies on a shear-free null hypersurface under integral curvature conditions.
method Using Minkowski formulas with arbitrary weight to derive rigidity results for submanifolds with weaker integral curvature conditions.
result A necessary and sufficient condition for a submanifold to lie in a shear-free null hypersurface is given by a mean curvature integral inequality.