Study on singularity behavior of mean curvature flow with bounded curvature and index.
problem Understanding singularity formation in mean curvature flow with constraints.
method Analyzing flow with bounded mean curvature and Morse index.
result Either mean curvature or Morse index blows up at first singular time.
New study confirms some mean curvature flow solutions have bounded mean curvature.
problem Existence of mean curvature flow singularities with bounded mean curvature.
method Construction of specific solutions in RN for N≥8. result A nontrivial subset of solutions has uniformly bounded mean curvature.
Upper bounds on area for surfaces with constant mean curvature in hyperbolic 3-manifolds.
problem Finding area constraints for surfaces with constant mean curvature in hyperbolic 3-manifolds.
method Established an upper bound on the area of closed embedded surfaces with constant mean curvature at least one, depending on the mean curvature and genus bounds.
result Area bound implies compactness for such surfaces, with specific proportional bounds for Bryant surfaces.
Compact mean curvature flow solutions with bounded curvature in high dimensions are constructed.
problem Constructing compact mean curvature flow solutions with bounded mean curvature.
method Following Velázquez, Guo, Sesum, and Stolarski's arguments, constructing solutions in \(\mathbb{R}^n\) with \(n \geq 8\).
result Compact mean curvature flow solutions with bounded mean curvature in \(\mathbb{R}^n\) are constructed.
Study shows unique tangent flow for Lagrangian surfaces with bounded mean curvature.
problem Understanding the behavior of Lagrangian surfaces with bounded mean curvature.
method Analyzing zero Maslov Lagrangian mean curvature flow in C2 with bounded mean curvature. result The tangent flow at a singular point is unique if the mean curvature stays uniformly bounded.
We give lower bounds for the first Dirichilet eigenvalues for domains in submanifolds with locally bounded mean curvatures. These bounds depend on the injectivity radius, sectional curvature (upperbound) of the ambient space and on the mean curvature of the submanifold. For submanifolds fo Hadamard manifolds these lowe…
Extends diameter bounds for submanifolds with boundary and minor curvature restrictions.
problem Bounding the diameter of submanifolds with boundary and minor curvature restrictions.
method Applies bounds dependent on mean curvature and area to minimal, constant mean curvature, and prescribed mean curvature surfaces.
result Diameter bounds for submanifolds with boundary and minor curvature restrictions.
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.
Proves Hamilton's theorem using mean curvature flow.
problem Compactness of pinched hypersurfaces with bounded curvature.
method Mean curvature flow to prove Hamilton's theorem.
result Rigorous proof of Hamilton's theorem.
Estimates mean curvature flow with geometric bounds.
problem Controlling mean curvature flow dynamics.
method Pointwise estimate using initial geometry and jHAj bound.
result Extension theorem and blowup rate estimate of HA.
Mean curvature flow with uniform bounds on curvature and its gradient
problem Mean curvature flow
method Uniform bounds on curvature and its gradient
result Smooth extension past singular time
Sharp bounds on mean curvature and geodesic lengths in convex hypersurfaces.
problem Finding sharp bounds on total mean curvature of convex hypersurfaces.
method Sharp lower bounds for mean width and Birkhoff invariant, characterizing spheres.
result Generalization of Álvarez Paiva's result to convex hypersurfaces.
Smooth approximations bound dihedral angles of convex polytopes.
problem Bounding dihedral angles of convex polytopes.
method Approximating polytopes with smooth hypersurfaces and using geometric relations.
result Established lower bounds on dihedral angles.
In this paper we prove that stable, compact without boundary, oriented, nonzero constant mean curvature surfaces in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds are the slices, provided its mean curvature satisfies some positive lower bound. More generally, we prove that stable, compact without boundary…
The study examines the graphical mean curvature flow on compact manifolds with bounded bi-Ricci curvature.
problem Analyzing the graphical mean curvature flow of maps between manifolds with bounded bi-Ricci curvature.
method Proving long-time existence and preserving the strictly area decreasing property under bounded bi-Ricci curvature conditions.
result Smooth convergence to a minimal map under certain conditions on Ricci 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.
In this paper we prove a compactness theorem for constant mean curvature surfaces with area and genus bound in three manifold with positive Ricci curvature. As an application, we give a lower bound of first eigenvalue of constant mean curvature surfaces in three manifold with positive Ricci curvature.
Study submanifolds in curved spaces with specific curvature bounds.
problem Investigate submanifolds in curved spaces with Ricci pinched conditions.
method Analyze submanifolds in Riemannian space forms with a lower Ricci curvature bound.
result Eliminate the need for mean curvature vector field to be parallel.
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.
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…
We prove a chord arc bound for disks embedded in R3 with constant mean curvature. This bound does not depend on the value of the mean curvature. It is inspired by and generalizes the work of Colding and Minicozzi in [2] for embedded minimal disks. Like in the minimal case, this chord arc bound is a fundamen…
Study shows uniform decay rate for singular mean curvature flows.
problem Understanding singularities in mean curvature flows.
method Rescaled flow analysis near compact singularities.
result Uniform decay order bound for the rescaled flow.
Sharp lower bound found for integral varifolds' mean curvature.
problem Finding a sharp lower bound for the mean curvature integral of integral varifolds.
method Developed a new approach using integral varifolds and mean curvature.
result A sharp lower bound on the mean curvature integral with critical power for integral varifolds.
3-manifolds with positive scalar curvature and bounded geometry are contractible.
problem Characterizing 3-manifolds with positive scalar curvature and bounded geometry.
method Maximal weak solution to inverse mean curvature flow.
result Complete contractible 3-manifolds with positive scalar curvature and bounded geometry are R3. The paper shows mean curvature flow keeps diameter bounded under certain conditions.
problem Proving the bounded diameter of hypersurfaces under mean curvature flow.
method Use of Lojasiewicz inequalities and solution of mean-convex neighbourhood conjecture.
result The intrinsic diameter stays uniformly bounded as the flow approaches the first singular time.
Upper bound for total mean curvature of spin fill-ins is proven.
problem Bounding the total mean curvature of spin Riemannian manifolds.
method Proving Gromov's conjecture for spin manifolds with specific conditions.
result Explicit upper bounds for total mean curvature are derived under various conditions.
In this paper we deal with singular varieties of bounded mean curvature in the viscosity sense. They contain all varifolds of bounded generalized mean curvature. In the first part we investigate the second-order properties of these varieties, obtaining results that are new also in the varifold's setting. In particular …
We study the provenance of singularity formation under mean curvature flow and volume preserving mean curvature flow in an axially symmetric setting. We prove that if the mean curvature is uniformly bounded on any finite time interval, then no singularities can develop during that time under both mean curvature flow an…
Self-shrinkers model singularities of the mean curvature flow; they are defined as the special solutions that contract homothetically under the flow. Colding-Ilmanen-Minicozzi showed that cylindrical self-shrinkers are rigid in a strong sense - that is, any self-shrinker that is mean convex with uniformly bounded curva…
In this note we consider asymptotically flat manifolds with non-negative scalar curvature and an inner boundary which is an outermost minimal surface. We show that there exists an upper bound on the mean curvature of a constant mean curvature surface homologous to a subset of the interior boundary components. This boun…
New bounds on genus and area for CMC surfaces in 3-manifolds.
problem Bounding genus and area of CMC surfaces in 3-manifolds.
method Local degeneration of minimal surfaces and index-area bounds.
result Genus and area of CMC surfaces are bounded by index and area.
Jorge-Koutrofiotis and Pigola-Rigoli-Setti proved sharp sectional curvature estimates for extrinsically bounded submanifolds. Alias, Bessa and Montenegro showed that these estimates hold on properly immersed cylindrically bounded submanifolds. On the other hand, Alias, Bessa and Dajczer proved sharp mean curvature esti…
Ancient curve shortening flows have entropy and curvature bounds equivalent.
problem Bounding entropy and total curvature for ancient curve shortening flows.
method Equivalence of entropy and total curvature conditions for ancient curve shortening flows.
result Entropy and total curvature bounds are equivalent for ancient curve shortening flows.
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.
Study bounds total mean curvature of fill-ins with scalar curvature constraints.
problem Bounding total mean curvature of fill-ins with scalar curvature constraints.
method Combines techniques from Shi-Tam, Shi-Wang-Wei, and recent work on systolic inequality.
result Sharp constant for total mean curvature estimate when boundary metric is flat.
Derives Hessian estimates for Lagrangian mean curvature equation.
problem Lagrangian mean curvature equation with supercritical phase and bounded second derivatives.
method Derives a priori interior Hessian estimates.
result Hessian estimates for Lagrangian mean curvature equation.
Paper relaxes convexity assumptions in mean curvature flow results.
problem Relaxing convexity assumptions in mean curvature flow results.
method Proves a generalized Harnack inequality and uses maximum principle.
result Characterizes family of shrinking spheres for ancient solutions.
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.
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. 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.
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 shows distance to boundary is always attained on varifolds with bounded curvature.
problem Understanding varifolds with bounded mean curvature in Riemannian manifolds.
method Proves a barrier principle at infinity using sharp maximum principles.
result Distance to boundary is always attained on varifolds with bounded curvature.
The paper proves bounds on mean curvature for CMC foliations with Ricci curvature constraints.
problem Bounding mean curvature for CMC foliations with specific Ricci curvature conditions.
method Analyzing foliations on compact Riemannian manifolds with Ricci curvature constraints.
result Proves that for a foliation by CMC hypersurfaces with specific Ricci curvature constraints, the mean curvature is constant and all leaves are totally umbilical.
Solves a specific Dirichlet problem for Lagrangian mean curvature equations.
problem Solving the Dirichlet problem for Lagrangian mean curvature equations.
method Solves the Dirichlet problem for Lagrangian mean curvature equations on uniformly convex domains.
result Solves the Dirichlet problem for Lagrangian mean curvature equations.
Unified estimates for mean curvature in Lorentz-Minkowski space.
problem Estimating mean curvature for space-like and time-like graphs.
method Using gradient bounds to derive Heinz-type estimates.
result Unified vanishing theorem for mean curvature of constant mean curvature graphs.
Sharp upper bounds on inscribed radius for metric spaces with convex boundary.
problem Bounding inscribed radius in metric measure spaces with convex boundary.
method Proves sharp upper bounds on inscribed radius for subsets with convex boundary.
result Sharp upper bounds on inscribed radius for subsets with convex boundary.
We consider a complete biharmonic hypersurface with nowhere zero mean curvature vector field φ:(Mm,g)→(Sm+1,h) in a sphere. If the squared norm of the second fundamental form B is bounded from above by m, and ∫MH−pdvg<∞, for some 0<p<∞, then the mean curvature is constant.