Smooth solutions found for modified mean curvature flow in Riemannian manifolds.
problem Existence of smooth solutions for modified mean curvature flow.
method A priori estimates for modified mean curvature flow in Riemannian manifolds with Killing vector field.
result Existence of smooth, entire, longtime solutions for modified mean curvature flow with smooth initial data.
The paper defines and studies the geometric mean for tensors and its associated Riemannian geometry.
problem Defining and studying the geometric mean for tensors.
method Generalized geometric mean for tensors using T-product, verified properties, and investigated Riemannian manifold.
result Geometric mean of T-positive definite tensors is a unique solution of algebraic Riccati tensor equations and a midpoint of geodesics.
Proves existence of sphere foliations with prescribed mean curvature on Riemannian manifolds.
problem Finding sphere foliations with prescribed mean curvature on Riemannian manifolds.
method Proves existence of foliations by spheres with mean curvature proportional to a given function on non-degenerate critical points.
result Essentially unique foliation of spheres with prescribed mean curvature exists in a neighborhood of a non-degenerate critical point.
Existence of double bubbles with high constant mean curvatures in Riemannian manifolds.
problem Existence of double bubbles with high constant mean curvatures in Riemannian manifolds.
method Perturbations of geodesic standard double bubbles centered at critical points of the ambient scalar curvature and aligned along eigen-vectors of the ambient Ricci tensor, with general multiplicity results via Lusternik-Schnirelman theory.
result Existence of double bubbles with high constant mean curvatures in Riemannian manifolds.
We develop some of the basic theory for the obstacle problem on Riemannian Manifolds, and we use it to establish a mean value theorem. Our mean value theorem works for a very wide class of Riemannian manifolds and has no weights at all within the integral.
Develops a duality for graphs in Riemannian and Lorentzian spaces with prescribed mean curvature.
problem Finding graphs with prescribed mean curvature in Riemannian and Lorentzian spaces.
method Introduces a conformal duality that swaps mean curvature and bundle curvature, invariant to base surface and reciprocal of the Killing vector field length.
result Entire graphs in Lorentz-Minkowski space with prescribed mean curvature a bounded function H.
GEORCE-FM algorithm optimizes Fréchet means and distances efficiently.
problem Computing Fréchet means on Riemannian manifolds efficiently.
method GEORCE-FM algorithm that simultaneously computes Fréchet means and distances in local charts.
result GEORCE-FM algorithm converges globally and locally quadratically, and scales to large datasets.
Study on stability of constant mean curvature hypersurfaces in Riemannian manifolds.
problem Stability of constant higher mean curvature hypersurfaces in Riemannian manifolds.
method Introduced a new notion of stability and used two stability operators to relate it to the first eigenvalues. Applied to Space Forms and proved non-stability for certain hypersurfaces.
result Embedded rotational spheres with constant k-mean curvature in HnxR or SnxR are not stable.
The paper examines 4D hypersurfaces with constant mean curvature in pseudo-Riemannian space forms.
problem Investigating properties of 4D hypersurfaces with specific curvature conditions.
method Analyzing hypersurfaces with proper mean curvature vector field in pseudo-Riemannian space forms.
result Bi-harmonic hypersurfaces in N^5_s(c) are minimal in certain cases.
New formulas compare total mean curvatures of nested hypersurfaces.
problem Computing total mean curvatures of nested hypersurfaces.
method Developed differential forms based on Chern's work to compare curvatures.
result Quicker proof of recent result on total mean curvatures.
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…
The paper derives inequalities for mean curvatures of hypersurfaces in Riemannian manifolds.
problem Geometric inequalities for mean curvatures of hypersurfaces in Riemannian manifolds.
method Comparison formula via Reilly's identities; geometric inequalities derived.
result Sharp lower bound for total first mean curvature in dimension 3.
Integrability of mean curvature near degenerate points in Heisenberg group.
problem Integrability of sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group.
method Introduction of mildly degenerate characteristic points and use of perimeter measure.
result The sub-Riemannian mean curvature is integrable in a neighborhood of these points.
Entropy defined for submanifolds; applies to mean curvature flow limits of surfaces.
problem Entropy for submanifolds in Riemannian manifolds.
method Entropy defined and shown to be monotone along mean curvature flow.
result Partial regularity of mean curvature flow limits of surfaces.
New examples of solitons found using submersion techniques.
problem Finding new mean curvature solitons on manifolds.
method Riemannian submersion techniques to reduce PDE to ODE.
result New examples of rotators in hyperbolic space.
We investigate the convergence of the mean curvature flow of arbitrary codimension in Riemannian manifolds with bounded geometry. We prove that if the initial submanifold satisfies a pinching condition, then along the mean curvature flow the submanifold contracts smoothly to a round point in finite time. As a consequen…
The study classifies parallel mean curvature spheres in a sphere-hyperbolic product space.
problem Understanding surfaces with parallel mean curvature in a specific Riemannian product space.
method Analyzing the holomorphic quadratic differential and topological constraints.
result Classification of all parallel mean curvature spheres with vanishing differential.
Study expands classical harmonic function results to Riemannian manifolds.
problem Classical harmonic function properties in domains of Riemannian manifolds.
method Generalized classical results to Riemannian manifolds, including pinched negative curvature.
result Generalized results for Riemannian manifolds, including pinched negative curvature.
Study shows how certain hypersurfaces evolve under mean curvature flow.
problem Evolution of isoparametric hypersurfaces under mean curvature flow.
method Reparametrization of the parallel family in short time.
result Evolution given by a reparametrization of the parallel family.
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.
We introduce a sub-Riemannian analogue of the Bence-Merriman-Osher diffusion driven algorithm and show that it leads to weak solutions of the horizontal mean curvature flow of graphs over sub-Riemannian Carnot groups. The proof follows the nonlinear semi-group theory approach originally introduced by L. C. Evans in the…
This study solves a Dirichlet problem for specific elliptic equations on Riemannian manifolds with concave boundaries.
problem Solving the Dirichlet problem for degenerate elliptic equations on Riemannian manifolds with mean concave boundaries.
method The proof relies on a quantitative boundary estimate.
result Analogous results are obtained in complex variables and on certain product manifolds.
Study on tautness tensor for Riemannian foliations.
problem Understanding tautness properties of Riemannian foliations.
method Investigating a symmetric 2-tensor related to mean curvature.
result Prove a tautness condition for compact manifolds.
Total curvatures of certain hypersurfaces are continuous.
problem Continuity of curvatures in geometric settings.
method Hausdorff distance for hypersurfaces and convex bodies in Riemannian manifolds and Cartan-Hadamard spaces.
result Total generalized mean curvatures are continuous.
Given a singular Riemannian foliation on a compact Riemannian manifold, we study the mean curvature flow equation with a regular leaf as initial datum. We prove that if the leaves are compact and the mean curvature vector field is basic, then any finite time singularity is a singular leaf, and the singularity is of typ…
Study on spectral properties of Riemannian submersions with special fibers.
problem Analyzing spectral properties of Riemannian submersions with fibers of basic mean curvature.
method Comparing the spectrum of the total space with a Schrödinger operator on the base manifold, extending results on Riemannian coverings.
result Computed the bottom of the spectrum and Cheeger constant for connected, amenable Lie groups.
It is proved the existence and uniqueness of graphs with prescribed mean curvature in Riemannian submersions fibered by flow lines of a vertical Killing vector field.
Study shows zero probability of cut locus for Fréchet mean on Riemannian manifolds.
problem Understanding the cut locus of Fréchet mean on Riemannian manifolds.
method Analytical proof and examples.
result Cut locus of Fréchet mean has zero probability.
We investigate the integral conditions to extend the mean curvature flow in a Riemannian manifold. We prove that the mean curvature flow solution with finite total mean curvature on a finite time interval [0,T) can be extended over time T. Moreover, we show that the condition is optimal in some sense.
Constructs ancient solutions to mean curvature flow with prescribed singular sets.
problem Creating mean-convex ancient solutions with specific singular sets.
method Constructs solutions with a prescribed singular set Kimes{0} using mean curvature flow in a Riemannian metric. result Constructs ancient solutions with a first-time singular set exactly Kimes{0}. We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isola…
Study on Riemannian foliations and their mean curvature components.
problem Understanding the mean curvature of Riemannian foliations.
method Analysis of the de Rham complex and basic cohomology.
result The basic component of mean curvature vanishes if and only if foliation leaves are minimal submanifolds.
Study shows no large mean curvature fill-ins for nonnegative scalar curvature.
problem Existence of fill-ins with nonnegative scalar curvature and large mean curvature.
method Analyzes closed Riemannian manifolds and scalar curvature properties.
result No closed Riemannian manifold admits a fill-in with nonnegative scalar curvature and large mean curvature.
3D metrics get scalar curvature bounds via IMCF.
problem Bounding scalar curvature for C0 metrics. method Inverse Mean Curvature Flow (IMCF) and Hawking mass monotonicity.
result Stability theorem for nonnegative scalar curvature.
The flag curvature of a Finsler metric is called a Riemannian quantity because it is an extension of sectional curvature in Riemannian geometry. In Finsler geometry, there are several non-Riemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the S-curvature, which all vanish for Ri…
Study phase transitions with prescribed mean curvature in Riemannian manifolds.
problem Understanding phase transitions with prescribed mean curvature in geometric settings.
method Analyzing solutions to inhomogeneous semilinear elliptic PDEs, establishing bounds and asymptotics.
result Established upper and lower bounds for eigenvalues of phase transition problems.
Given an m-dimensional closed connected Riemannian manifold M smoothly isometrically immersed in an n-dimensional Riemannian manifold N, we estimate the diameter of M in terms of its mean curvature field integral under some geometric restrictions, and therefore generalize a recent work of Topping in the Eucli…
We consider the product of a compact Riemannian manifold without boundary and null scalar curvature with a compact Riemannian manifold with boundary, null scalar curvature and constant mean curvature on the boundary. We use bifurcation theory to prove the existence of a infinite number of conformal classes with at leas…
Study duality of zero mean curvature surfaces in Heisenberg group.
problem Understanding the duality of zero mean curvature surfaces in the Lorentzian Heisenberg group.
method Investigation of a transformation surface associated with zero mean curvature surfaces in the Heisenberg group under two metrics.
result Derivation of the Sym formula for the dual surface in both metric cases.
We show that mean curvature flow of a compact submanifold in a complete Riemannian manifold cannot form singularity at time infinity if the ambient Riemannian manifold has bounded geometry and satisfies certain curvature and volume growth conditions .
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.
We give a sufficient condition ensuring that the mean curvature flow commutes with a Riemannian submersion and we use this result to create new examples of evolution by mean curvature flow. In particular we consider evolution of pinched submanifolds of the sphere, of the complex projective space, of the Heisenberg grou…
Proves uniqueness of geometric flow in various Riemannian manifolds.
problem Proving uniqueness of geometric flow in general Riemannian manifolds.
method Two backward uniqueness theorems for extrinsic geometric flow.
result Backward uniqueness of extrinsic geometric flow in general ambient manifolds.
Paper provides lower bounds for eigenvalues on singular Riemannian foliations.
problem Lower bounds for the first non-zero basic eigenvalue on singular Riemannian manifolds.
method Generalized Zhong-Yang and Shi-Yang estimates for singular Riemannian foliations with basic mean curvature.
result Rigidity result when the first basic eigenvalue equals a specific value.
This paper is a short summary of our recent work on the medians and means of probability measures in Riemannian manifolds. Firstly, the existence and uniqueness results of local medians are given. In order to compute medians in practical cases, we propose a subgradient algorithm and prove its convergence. After that, F…
Solves a problem in Riemannian geometry for scalar-flat metrics with boundary conditions.
problem Finding a conformal metric with zero scalar curvature and prescribed boundary mean curvature.
method Construction of local test functions to resolve open cases and establish new solvability conditions.
result Established new solvability conditions for the problem.
This is a survey of our work on spacelike graphic submanifolds in pseudo-Riemannian products, namely on Heinz-Chern and Bernstein-Calabi results and on the mean curvature flow, with applications to the homotopy of maps between Riemannian manifolds.
Study on ion travel time on curved surfaces.
problem Mean first passage time of ion on curved surfaces.
method Layer potential argument and microlocal analysis.
result Derivation of mean first passage time and spatial average.