The paper constructs surfaces with prescribed mean curvature in a specific space.
problem Finding surfaces with a given mean curvature in a particular geometric space.
method Phase plane analysis to construct entire rotational graphs and catenoid-type surfaces.
result Classification result for surfaces with linearly prescribed mean curvature.
Paper proves Harnack inequality for f-mean curvature flow.
problem Proving Harnack inequality for f-mean curvature flow. method Gradient flow of the weighed area functional with measure density function e−f. result Proves Li-Yau-Hamilton type Harnack estimate.
The paper explores rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space.
problem Rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space.
method Characterizations and rigidity investigations for hypersurfaces with constant weighted shifted mean curvatures or ratios.
result Rigidity results for hypersurfaces with constant linear combinations of weighted shifted mean curvatures and radially symmetric shifted mean curvatures.
Classification of surfaces with prescribed mean curvature in Heisenberg space and SL2(R).
problem Surfaces with prescribed mean curvature in Heisenberg space and SL2(R).
method Classification result for rotational surfaces with prescribed mean curvature.
result Existence of embedded tori as counterexamples to the Alexandrov problem.
In this paper, we shall prove that space-like surfaces with bounded mean curvature functions in real analytic Lorentzian 3-manifolds can change their causality to time-like surfaces only if the mean curvature functions tend to zero. Moreover, we shall show the existence of such surfaces with non-vanishing mean curvatur…
Proves optimal regularity for sphere minimizers in 3-sphere.
problem Finding optimal regularity for sphere minimizers.
method Proves C1,1 regularity for minimizers of prescribed mean curvature over isotopy classes. result Proves optimal C1,1 regularity for minimizers. Compact capillary hypersurfaces with specific curvatures are proven.
problem Proving compactness of capillary hypersurfaces with prescribed mean curvature.
method Extending curvature varifolds to oriented integral varifolds and applying curvature varifolds with capillary boundary.
result Compactness result for capillary hypersurfaces with mean curvature prescribed by ambient functions.
We use a phase space analysis to give some classification results for rotational hypersurfaces in Rn+1 whose mean curvature is given as a prescribed function of its Gauss map. For the case where the prescribed function is an even function in Sn, we show that a Delaunay-type classification hold…
The study examines continuous mean curvature functions on manifolds without conjugate points.
problem Understanding properties of manifolds with specific curvature functions.
method Analyzing simply connected Riemannian manifolds with continuous horospherical mean curvature functions.
result Compact rank one manifolds without conjugate points are locally symmetric spaces of negative curvature.
Proves Arnold-Thom conjecture for surfaces' arrival times.
problem Existence of limit tangents for gradient flow lines of surfaces.
method Gradient flow lines of mean curvature flows with neck or cylindrical singularities.
result Proves Arnold's conjecture for all mean convex mean curvature flows of surfaces.
The study examines constant weighted mean curvature hypersurfaces in shrinking Ricci solitons.
problem Characterizing constant weighted mean curvature hypersurfaces in shrinking Ricci solitons.
method Analyzing properties of hypersurfaces in specific ambient spaces (shrinking Ricci solitons).
result Conditions for a constant weighted mean curvature hypersurface to be a level set of the potential function.
Proves existence of special 2-spheres in curved 3-spaces.
problem Existence of constant mean curvature 2-spheres in Riemannian 3-spheres.
method Develops a min-max scheme for a weighted Dirichlet energy functional, using bi-harmonic approximation, derivative estimates, and Morse index estimates.
result Proves existence for almost every mean curvature and all for positively curved 3-spheres.
Study Hamiltonian stationary Lagrangian surfaces in complex space forms.
problem Characterize Lagrangian surfaces with harmonic mean curvature in complex space forms.
method Analyze surfaces with constant and harmonic mean curvature, using second fundamental form parallelism and Gaussian curvature constancy.
result Complete classification of Lagrangian surfaces with harmonic mean curvature and constant Gaussian curvature.
Theory proves existence of hypersurfaces with prescribed curvature.
problem Existence of hypersurfaces with prescribed mean curvature in noncompact manifolds.
method Developed min-max theory for noncompact manifolds.
result Proved existence of closed and finite area hypersurfaces.
Study nondegenerate singularities in mean curvature flow.
problem Understanding the behavior of nondegenerate cylindrical singularities.
method New L2-distance monotonicity formula and discrete almost monotonicity. result Topology change agrees with level sets change near a critical point of a Morse function.
Proves existence and uniqueness of Killing graphs with prescribed curvature.
problem Existence and uniqueness of Killing graphs with prescribed curvature.
method Proves existence and uniqueness of Killing graphs with prescribed mean curvature considering non-constant functions.
result Existence and uniqueness of Killing graphs with prescribed curvature.
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.
We prove that, for a generic set of smooth prescription functions h on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature h. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersur…
This paper gives a new proof that maximal, globally hyperbolic, flat spacetimes of dimension n≥3 with compact Cauchy hypersurfaces are globally foliated by Cauchy hypersurfaces of constant mean curvature, and that such spacetimes admit a globally defined constant mean curvature time function precisely when they a…
Minimal surfaces in harmonic conformally flat space are studied.
problem Minimal surfaces in harmonic conformally flat space.
method Variational geometry approach focusing on mean curvature and Willmore functionals.
result Critical points of mean curvature functional are homeomorphic to the sphere.
Study self-expanding solutions of mean curvature flow in various dimensions.
problem Characterize complete mean convex self-expanding hypersurfaces and their properties.
method Analyzing the function ∣A∣2/∣H∣2 and ∣Aξ∣2/∣H∣2 to understand the structure of self-expanders. result Complete mean convex self-expanders are products of self-expanding curves and flat subspaces under certain conditions.
Compactness results for hypersurfaces with mean curvature prescribed by an ambient function.
problem Proving compactness for hypersurfaces with prescribed mean curvature.
method Using oriented integral varifolds and a weak notion of curvature coefficients.
result Locally uniform bounds on second fundamental form lead to compactness.
The paper classifies shapes of translating solitons for a specific flow.
problem Understanding the shapes of translating solitons in a specific flow.
method Analyzing functions on a unit sphere and solving an ODE.
result Classification of the shapes of translating solitons.
We prove that the mean curvature τ of the slices given by a constant mean curvature foliation can be used as a time function, i.e. τ is smooth with non-vanishing gradient.
The paper proves the monotonicity of a modified Perelman's W-entropy for mean curvature flow.
problem Proving the monotonicity of Perelman's W-entropy for mean curvature flow.
method Modified definition of K. Ecker's W-entropy and Hamilton's Harnack inequality.
result The modified W-entropy is monotonically decreasing in time.
Study finds first integrals in Finsler metrics with vanishing χ-curvature.
problem Analyzing Finsler metrics with vanishing χ-curvature.
method Proving geodesic invariance of certain structures and deriving first integrals.
result Induces a set of first integrals in Finsler metrics with vanishing χ-curvature.
Study shows bound on Uryson width for specific 3D manifolds.
problem Bounding Uryson width for 3D manifolds with non-negative Ricci curvature and strictly mean convex boundary.
method Proved existence of a Morse function with uniform diameter bounds on level sets.
result Upper bound on Uryson width for the specified 3D manifolds.
Study of mean curvature flow in warped products preserving equivariance.
problem Analyzing mean curvature flow in warped products.
method Deriving flow equation and proving existence for infinite time.
result Mean curvature flow exists for infinite time under specific conditions.
Constructs hyperspheres with prescribed mean curvature in Euclidean space.
problem Creating hyperspheres with a specific curvature in Euclidean space.
method Constructs families of smooth functions to fill Euclidean space with hyperspheres of prescribed mean curvature.
result Euclidean space can be filled with hyperspheres of prescribed mean curvature.
Two spheres found with specific curvature constraints.
problem Existence of spheres with prescribed mean curvature.
method Proved existence of at least two embedded spheres with curvature h satisfying pinching condition. result Existence of at least two embedded spheres with prescribed mean curvature h. A submanifold of a pseudo-Riemannian manifold is said to have parallel mean curvature vector if the mean curvature vector field H is parallel as a section of the normal bundle. Submanifolds with parallel mean curvature vector are important since they are critical points of some natural functionals. In this paper, we su…
We obtain the explicit representation of Legendre surfaces in the unit 5-sphere with harmonic mean curvature vector field, under the condition that the mean curvature function is constant along a certain special direction.
Study of spacetimes in cosmology without symmetry assumptions.
problem Proving timelike incompleteness and properties of time functions.
method Rigorous mathematical proofs and analysis of spacetime properties.
result Timelike incompleteness for specific spacetimes with mean curvature constraints.
Two-dimensional Lagrangian mean curvature equation solved with new inequality.
problem Solving the two-dimensional Lagrangian mean curvature equation.
method Using Warren-Yuan's super isoperimetric inequality and simplified approach.
result Derivation of a modified Hessian bound for solutions.
Estimates Bartnik mass for metrics with nonnegative Gauss curvature.
problem Estimating Bartnik mass for specific metric configurations.
method Using area, total mean curvature, and a metric roundness measure.
result Estimate approaches sharp value for round spheres.
Local classification of surfaces and hypersurfaces with radial mean curvature.
problem Classifying surfaces and hypersurfaces with specific curvature properties.
method Local classification and recursive construction method.
result Local classification of hypersurfaces with vanishing second mean curvature.
We extend the interior gradient estimate due to N. Korevaar and L. Simon for solutions of the mean curvature equation from the case of Euclidean graphs to the general case of Killing graphs. Our main application is the proof of existence of Killing graphs with prescribed mean curvature function for continuous boundary …
This paper concerns the evolution of a closed convex hypersurface in Rn+1, in direction of its inner unit normal vector, where the speed is given by a smooth function depending only on the mean curvature, and satisfies some further restrictions, without requiring homogeneity. It is shown that the flow e…
Proves curvature comparison for Riemannian bands in low dimensions.
problem Curvature comparison in Riemannian bands with lower bounds.
method Uses warped products over scalar-flat manifolds with log-concave warping.
result Scalar and mean curvature comparison results proven.
In this note we study a large class of mean curvature type flows of graphs in product manifold N×R where N is a closed Riemann- ian manifold. Their speeds are the mean curvature of graphs plus a prescribed function. We establish long time existence and uniformly convergence of those flows with a barrier conditi…
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, …
The purpose of this paper is to study immersed surfaces in the product spaces M2(κ)×R, whose mean curvature is given as a C1 function depending on their angle function. This class of surfaces extends widely, among others, the well-known theory of surfaces with constant mean curvature. In th…
We show that under suitable non-degeneracy conditions, complete gradient flow lines of the scalar curvature functional of a riemannian manifold perturb into eternal forced mean curvature flows with large forcing term.
In this paper, we derive the evolution equation for the first eigenvalue of the Witten-Laplace operator acting on the space of functions along the mean curvature flow on a closed oriented manifold. We show some interesting monotonic quantities under the mean curvature flow.
Using the convex functions in Grassmannian manifolds we can carry out interior estimates for mean curvature flow of higher codimension. In this way some of the results of Ecker-Huisken can be generalized to higher codimension
New existence results for curvature problem on balls with specific conditions.
problem Existence of solutions for a prescribed mean curvature problem on a ball.
method Combining critical points at infinity approach with Morse theory.
result New existence results for higher dimensional case n≥5 under pinching conditions. The study extends curvature bounds to non-smooth spaces and proves stability of mean curvature.
problem Proving curvature bounds in non-smooth spaces.
method Extending results from smooth Riemannian manifolds to non-smooth RCD spaces.
result Stability of mean curvature bounds under uniform convergence.
We show that closed hypersurfaces in Euclidean space with nonnegative scalar curvature are weakly mean convex. In contrast, the statement is no longer true if the scalar curvature is replaced by the k-th mean curvature, for k greater than 2, as we construct the counter-examples for all k greater than 2. Our proof relie…