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.
We show that strictly convex surfaces expanding by the inverse Gauss curvature flow converge to infinity in finite time. After appropriate rescaling, they converge to spheres. We describe the algorithm to find our main test function.
By means of dual convex bodies, we obtain regularity of solutions to the expanding Gauss curvature flows with homogeneity degrees −p, 0<p<1. At the end, we remark that our method can also be used to obtain regularity of solutions to the shrinking Gauss curvature flows with homogeneity degrees less than one.
Study anisotropic inverse Gauss curvature flows and solve dual Orlicz Minkowski problems.
problem Solving dual Orlicz Minkowski problems for anisotropic flows.
method Anisotropic inverse Gauss curvature flows and stationary solutions.
result New existence results for dual Orlicz Minkowski problems for smooth measures.
The paper studies how spacelike surfaces evolve in Lorentz-Minkowski space over time.
problem Evolution of spacelike graphic hypersurfaces in Lorentz-Minkowski space.
method Inverse Gauss curvature flow with Neumann boundary condition.
result The evolving surfaces converge to a constant function as time goes to infinity.
Unified view of monotonicity formulas for inverse mean curvature flow and p-capacitary potentials.
problem Understanding monotonicity formulas for various geometric flows and potentials.
method Refined analysis of p-capacitary potentials and their level sets. result Strong convergence of p-capacitary potentials to inverse mean curvature flow and curvature varifolds. The paper explores Kα-translators on parallel and canal surfaces in 3D space.
problem Investigating conditions for Kα-translators on parallel and canal surfaces. method Analyzing the conditions for Kα-translators on parallel surfaces and canal surfaces, proving their properties and existence. result No Kα-translators exist on the parallel surface of a rotational surface obtained from a canal surface with the same speed w, while the rotational surface itself is a Kα-translator. We consider strictly convex hypersurfaces with the boundary which meets a strictly convex cone perpendicularly. We prove that if these hypersurfaces expand inside this cone, driven by the power of the Gauss curvature, then the evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece…
We prove gradient estimates for hypersurfaces in the hyperbolic space Hn+1, expanding by negative powers of a certain class of homogeneous curvature functions. We obtain optimal gradient estimates for hypersurfaces evolving by certain powers p>1 of F−1 and smooth convergence of the properly rescale…
Proves existence of unique circle packings on polyhedral surfaces.
problem Existence of unique circle packings on polyhedral surfaces with specified discrete curvature.
method Constructs diffeomorphism between fiber bundles, uses discrete Ricci flow and edge flipping.
result Proves existence of unique inversive distance circle packings.
We prove a rigidity result in the sphere which allows us to generalize a result about smooth convex hypersurfaces in the sphere by Do Carmo-Warner to convex C2-hypersurfaces. We apply these results to prove C1,β-convergence of inverse F-curvature flows in the sphere to an equator in \mathbb{S}^{n+1} for embedde…
We define a formal Riemannian metric on a given conformal class of metrics on a closed Riemann surface. We show interesting formal properties for this metric, in particular the curvature is nonpositive and the Liouville energy is geodesically convex. The geodesic equation for this metric corresponds to a degenerate ell…
Paper solves Orlicz-Aleksandrov problem using Gauss curvature flow.
problem Orlicz-Aleksandrov problem
method Gauss curvature flow
result Existence of smooth solutions
In this paper, we investigate the Gauss maps of a Ricci-mean curvature flow. A Ricci-mean curvature flow is a coupled equation of a mean curvature flow and a Ricci flow on the ambient manifold. Ruh and Vilms proved that the Gauss map of a minimal submanifold in a Euclidean space is a harmonic map, and Wang extended thi…
The paper constructs ancient solutions to curvature flows in bounded and unbounded regions.
problem Understanding ancient solutions to curvature flows in bounded and unbounded regions.
method Constructing pancake-like and sausage-like ancient compact solutions.
result Ancient solutions to curvature flows in bounded and unbounded regions.
Study examines noncompact cases of Gauss Curvature Flow on revolution surfaces.
problem Noncompact cases of Gauss Curvature Flow on revolution surfaces.
method Examines two noncompact cases of Gauss Curvature Flow on revolution surfaces.
result Examines noncompact cases of Gauss Curvature Flow on revolution surfaces.
The paper proves convergence of certain curvature flows to the origin.
problem Analyzing the convergence of specific curvature flows in Euclidean space.
method Examining fully nonlinear contracting curvature flows with given normal speeds.
result The flows converge exponentially to a sphere centered at the origin after rescaling.
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.
Study on the regularity of p-Gauss curvature flow near flat interfaces.
problem Regularity of p-Gauss curvature flow near flat interfaces. method Analysis of convex hypersurface near the interface.
result Regularity of the convex hypersurface near the interface.
Study of capillary Gauss curvature flow shrinking to a point.
problem Finite time shrinkage of capillary hypersurfaces.
method Introduced a Gauss curvature type flow for capillary hypersurfaces.
result Normalized flow converges to a soliton.
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 c…
Flow adjusts curvature to avoid a fixed region, proving bounds and regularity.
problem Adjusting curvature flow to avoid a fixed region.
method Flow by powers of Gauss curvature, proving optimal curvature bounds and regularity.
result Proves optimal curvature bounds and long time existence for all dimensions and powers.
The paper studies how convex hypersurfaces evolve under curvature flows in space forms.
problem Understanding the evolution of convex hypersurfaces under curvature flows in different space forms.
method Flow by powers of the Gauss curvature in space forms.
result Convex hypersurfaces under the flow by powers of the Gauss curvature in space forms contract to a point in finite time or converge to geodesic spheres.
Study of a flow related to the Orlicz-Minkowski problem for convex hypersurfaces.
problem Orlicz-Minkowski problem involving Gauss curvature and support function.
method Generalized Gauss curvature flow for convex hypersurfaces in Euclidean n-space.
result Long-time existence and convergence of the flow, leading to existence results for the Orlicz-Minkowski problem.
We study the mean curvature flow of complete space-like submanifolds in pseudo-Euclidean space with bounded Gauss image, as well as that of complete submanifolds in Euclidean space with convex Gauss image. By using the confinable property of the Gauss image under the mean curvature flow we prove the long time existence…
Using polar convex bodies and the C0-bounds from Guan and Ni \cite{PL}, we obtain a uniform lower bound on the Gauss curvature of the normalized solution of the Gauss curvature flow without using Chow's Harnack inequality \cite{Ch2}.
Study translating solitons in Minkowski space with prescribed Gauss image.
problem Second boundary value problem for mean curvature flow in Minkowski space.
method Construct translating solitons with prescribed Gauss image.
result Constructing translating solitons with prescribed Gauss image in Minkowski space.
Two ancient solutions to Gauss curvature flow are identified for cylinders.
problem Classifying ancient solutions to Gauss curvature flow in cylinders.
method Assumption of cylinder cross-section bounded convexity, analysis of asymptotic behavior.
result Only two ancient solutions identified: translating soliton and compact oval solution.
The paper classifies invariant translators for a specific curvature flow.
problem Classifying invariant translators for a specific curvature flow.
method Classification of λ-translators invariant under translations and rotations. result All λ-translators are classified. The paper proves the existence of a unique circle packing on hyperbolic surfaces.
problem Proving the existence of a unique inversive distance circle packing on hyperbolic polyhedral surfaces.
method Deforming the surface by discrete Ricci flow, doing surgery by edge flipping, and using a variational principle of a convex Ricci potential.
result There exists a unique inversive distance circle packing that is discrete conformal to the original one.
Study curve shortening flows on specific surfaces, proving properties and existence.
problem Analyzing curve shortening flows on rotational surfaces with negative Gauss curvatures.
method Assume negative Gauss curvatures and conditions on Gauss curvature and curve curvature. Prove curve remains a graph and establish flow properties.
result Prove the curve remains a graph over parallels and establish long-time existence of the flow.
The paper constructs hypersurfaces translating under powers of Gauss curvature.
problem Existence of hypersurfaces translating under powers of Gauss curvature.
method Constructs complete convex hypersurfaces in R^(n+1) translating under flow by powers of Gauss curvature.
result Existence of translators whose level set converges to various shapes like sphere, simplex, and hypercube.
The paper studies how convex hypersurfaces in hyperbolic space evolve under a specific curvature flow.
problem Volume preserving Gauss curvature flow in hyperbolic space.
method Analyzes a flow of smooth, closed, and convex hypersurfaces in hyperbolic space with a nonhomogeneous speed function.
result The flow remains convex, exists for all time, and converges to a geodesic sphere exponentially.
The paper classifies special solitons and shrinkers in Euclidean space.
problem Characterizing special solitons and shrinkers in Euclidean space.
method Analyzing λ-translating solitons and λ-shrinkers with constant mean curvature. result Planes, spheres, and circular cylinders are the only λ-shrinkers and λ-translating solitons with constant mean curvature. We consider flows with normal velocities equal to powers strictly larger than one of the Gauss curvature. Under such flows closed strictly convex surfaces converge to points. In his work on the square of the norm of the second fundamental form, Schnürer proposes criteria for selecting quantities that are suitable for p…
Smooth solutions up to evolving free boundaries for degenerate equations.
problem Degenerate parabolic equations with evolving free boundaries.
method Smooth short-time existence using linear degenerate equations on a fixed domain.
result Smoothness up to the free boundary for the p-Laplacian evolution equation and α-Gauss curvature flow. Paper solves Minkowski problem for p-harmonic measures.
problem Solving the Minkowski problem for p-harmonic measures on convex domains.
method Using the Gauss curvature flow method.
result Existence of smooth solution to the Minkowski problem for p-harmonic measures.
Flow of convex hypersurfaces in hyperbolic space converges to geodesic spheres.
problem Understanding the evolution of convex hypersurfaces in hyperbolic space.
method Gauss curvature type flow, Alexandrov-Fenchel inequality application.
result Smooth solution converges to geodesic spheres.
New method solves a generalized Minkowski problem using a curvature flow.
problem Generalized Minkowski problem for smooth measures.
method Flow involving Gauss curvature and support function.
result Existence of solutions for the dual Orlicz-Minkowski problem.
Study on anisotropic curvature flow for noncompact convex hypersurfaces.
problem Anisotropic curvature flow of noncompact convex hypersurfaces.
method Flow of complete noncompact convex hypersurfaces with anisotropy determined by a Wulff shape.
result The flow exists for all positive time for initial conditions.
Proves long-term smoothness of curved surfaces evolving under specific curvature rules.
problem Long-term regularity of curved surfaces evolving under p-Gauss curvature flow. method Transformed the curvature flow into a Monge-Ampère equation and studied its asymptotic cone.
result Proved regularity of the interface in all dimensions for $p> rac1n$.
We prove the asymptotic roundness under normalized Gauss curvature flow provided entropy is initially small enough.
Solves Lp-Gaussian chord Minkowski problem using Gauss curvature flow.
problem Solving the Lp-Gaussian chord Minkowski problem. method Using Gauss curvature flow to obtain smooth even solutions.
result Obtains smooth even solutions to the Lp-Gaussian chord Minkowski problem. The skew mean curvature flow (SMCF) is a natural generalization of the famous vortex filament equation. In this note, we show that the Gauss map of the SMCF satisfies a Schrödinger flow equation. In this regard, we explore the geometry of the oriented Grassmannian manifold explicitly by embedding it into the exterior p…
Ginger efficiently approximates curvature with linear complexity for neural networks.
problem Quadratic memory and cubic time complexity for computing curvature matrices in deep learning.
method Ginger uses eigendecomposition to maintain the inverse of the generalized Gauss-Newton matrix, achieving linear memory and time complexity.
result Ginger provides an effective and efficient curvature approximation for non-convex objectives.
Proves existence of proper solutions for inverse mean curvature flow.
problem Existence of proper solutions for inverse mean curvature flow.
method Proves existence theorem assuming non-degeneracy conditions on isoperimetric profile.
result No curvature assumption in existence theorem.
Classifies surfaces translating under specific curvature flows.
problem Classifying surfaces translating under flows by sub-affine-critical powers of Gauss curvature.
method Analyzes entire graphs of surfaces translating under flows by sub-affine-critical powers of the Gauss curvature.
result Lists all translating solitons possibly model Type II singularities for convex closed solutions in all positive powers.
The paper studies how surfaces move by mean curvature flow and what happens at singular points.
problem Understanding the behavior of surfaces moving by mean curvature flow at singular points.
method Proves that tangent flows at singular times are smooth shrinkers, with a new local Gauss-Bonnet formula.
result Smooth shrinkers without branch points if the initial surface is embedded in 3-manifold.