The paper studies the blow-up of conformal mean curvature flow in higher codimension.
problem Analyzing the blow-up behavior of conformal mean curvature flow.
method Introduced and studied conformal mean curvature flow, derived blow-up theorem and evolution formulas.
result Maximum of the square norm of the second fundamental form tends to infinity in finite time.
Article explains and implements mean curvature flow for surface parametrization.
problem Surface parametrization challenges.
method Conformalized mean curvature flow implementation.
result Demonstrates effectiveness of mean curvature flow for surface parametrization.
The round sphere is the only solution to inverse curvature flows under certain conditions.
problem Proving uniqueness of self-conformal solutions to inverse curvature flows.
method Analyzing flows by diffeomorphisms generated by conformal Killing fields.
result The round sphere is the only closed solution to the inverse mean curvature flow and related flows under natural conditions.
We employ three different methods to prove the following result on prescribed scalar curvature plus mean curvature problem: Let (Mn,g0) be a n-dimensional smooth compact manifold with boundary, where n≥3, assume the conformal invariant Y(M,∂M)<0. Given any negative smooth functions f in M and…
Study proves mean curvature flow in GRW spacetimes with perpendicular boundary condition.
problem Longtime existence of mean curvature flow in GRW spacetimes.
method Proved longtime existence using perpendicular Neumann boundary condition and null convergence condition.
result Metric of solution is conformal to GRW leaf's metric in asymptotic time.
The paper defines solitons for mean curvature flow on curved spaces.
problem Mean curvature flow on curved spaces.
method Introduces a general definition of solitons for mean curvature flow.
result Defines solitons for a broader class of spaces.
The paper classifies solitons for mean curvature flow in hyperbolic space.
problem Mean curvature flow in hyperbolic space.
method Study of conformal solitons in the upper half-space model of hyperbolic space.
result Classification of cylindrical and rotationally symmetric examples, including grim-reaper cylinders and bowl/winglike solitons.
Study proves existence of curvature-preserving metrics on balls with symmetries.
problem Existence of conformal metrics with prescribed mean curvature on balls.
method Flow method to prove existence of solutions.
result Existence of conformal metrics with prescribed mean curvature on unit balls with symmetries.
Paper extends isoperimetric inequalities for non-starshaped hypersurfaces.
problem Isoperimetric inequalities for non-starshaped hypersurfaces.
method Volume preserving and area decreasing mean curvature flow with conformal Killing vector fields.
result Established isoperimetric inequalities for a broader class of hypersurfaces.
Study of IMCF on warped product manifolds for long time existence.
problem Analyzing IMCF on conformal warped product manifolds.
method Relating gradient conformal vector field to conformal vector field, using control of the flow.
result Showed long time existence of IMCF on conformal warped product manifolds.
Study inverse mean curvature flow in quaternionic hyperbolic space, proving flow properties and convergence.
problem Evolution of star-shaped hypersurfaces in quaternionic hyperbolic space.
method Inverse mean curvature flow, star-shaped hypersurface, mean convex, convergence analysis.
result Flow is defined for any positive time, evolving hypersurface stays star-shaped and mean convex, induced metric converges to a conformal multiple of the standard sub-Riemannian metric on the sphere.
Study the isoperimetric problem in Riemannian manifolds with non-trivial conformal vector fields.
problem Isoperimetric problem in Riemannian manifolds with non-trivial conformal vector fields.
method Introduced a mean curvature type flow to study the isoperimetric problem.
result Established the isoperimetric inequality for star-shaped hypersurfaces in such manifolds.
Given a compact four dimensional smooth Riemannian manifold (M,g) with smooth boundary, we consider the evolution equation by Q-curvature in the interior keeping the T-curvature and the mean curvature to be zero and the evolution equation by T-curvature at the boundary with the condition that the Q-curvature …
We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the complex hyperbolic space. We prove that the flow is defined for any positive time, the evolving hypersurface stays star-shaped and mean convex. Moreover the induced metric converges, after rescaling, to…
Study on curvature conditions for non-conformally flat spheres using quasiconformal maps and Ricci flow.
problem Curvature conditions for non-conformally flat spheres.
method Construct quasiconformal maps and apply Ricci flow.
result Controlled bilipschitz constant between metrics.
The mean curvature flow describes the parabolic deformation of embedded branes in Riemannian geometry driven by their extrinsic mean curvature vector, which is typically associated to surface tension forces. It is the gradient flow of the area functional, and, as such, it is naturally identified with the boundary renor…
Paper solves curvature prescription on unit ball with sign-changing functions.
problem Prescribing mean curvature on the unit ball with sign-changing functions.
method Negative gradient flow method to realize f as mean curvature. result Proves that a sign-changing function f can be realized as the boundary mean curvature of a conformal metric. Study classifies minimal surfaces and solitons in hyperbolic 3-space as translation surfaces.
problem Classifying minimal surfaces and solitons in hyperbolic 3-space.
method Investigates minimal surfaces and solitons to the mean curvature flow in hyperbolic three-space using specific product forms of curves.
result Provides classification results for minimal surfaces, hyperbolic translators, and conformal solitons.
Entropy for submanifolds in hyperbolic space defined.
problem Entropy for submanifolds in hyperbolic space.
method Entropy defined analogous to Euclidean space.
result Entropy monotonicity along mean curvature flow in low dimensions.
The study examines evolving star-shaped hypersurfaces in hyperbolic spaces, influenced by ambient geometry.
problem Evolution of star-shaped hypersurfaces in hyperbolic spaces.
method Nonhomogeneous expanding curvature flows in hyperbolic spaces.
result The asymptotic behavior of the flow depends on the ambient space's geometry, leading to different limiting metrics.
New perspective on Ricci flow on spheres using Minkowski spacetime.
problem Classifying singularity models for null mean curvature flow in Minkowski spacetime.
method Equivalence of 2d-Ricci flow and null mean curvature flow on lightcones.
result Classification of singularity models for null mean curvature flow.
Liouville entropy increases strictly along Ricci flow on surfaces.
problem Understanding the behavior of Liouville entropy under Ricci flow.
method New expression for Liouville entropy derivative, proof of positivity in specific directions.
result Liouville entropy is strictly increasing along normalized Ricci flow for 1/6-pinched metrics.
Study on metrics with positive scalar curvature and convex boundary.
problem Characterizing 3-manifolds with positive scalar curvature and convex boundary.
method Combination of earlier contributions, smoothing procedure, Ricci flow, and conformal deformation techniques.
result Path-connectedness of the moduli space of metrics.
The abstract discusses compactness of manifolds with pinched Ricci curvature.
problem Prove that a complete Riemannian manifold with positively pinched Ricci curvature is compact.
method Detailed alternate proof using quasi-conformal maps and mean curvature flow.
result Provides a proof of Hamilton's result on compactness of convex hypersurfaces.
Study self translating solitons in a product space, relating them to manifolds with density.
problem Understanding self translating solitons in H2imesR. method Relating to manifolds with density, constructing examples, studying asymptotic behavior, proving theorems.
result Proved some uniqueness and non-existence theorems.
The paper studies deformation of discrete conformal structures on surfaces using combinatorial curvature flows.
problem Finding piecewise constant curvature metrics on surfaces with prescribed combinatorial curvatures.
method Combinatorial curvature flows, including Ricci flow and Calabi flow, are applied to deform Glickenstein's discrete conformal structures.
result The solution of the combinatorial Ricci flow can be uniquely extended and converges exponentially fast for any initial value under certain conditions.
The paper studies how submanifolds in Gaussian space behave under mean curvature flow, showing they typically blow up.
problem Behavior of submanifolds in Gaussian space under mean curvature flow.
method Analysis of mean curvature flow in the standard Gaussian metric space.
result Submanifolds in Gaussian space with non-zero square norm of position vector blow up under mean curvature flow.
Study short-time existence of conformal Ricci flow on hyperbolic manifolds.
problem Short-time existence of conformal Ricci flow on asymptotically hyperbolic manifolds.
method Proved local Shi's type curvature derivative estimate for conformal Ricci flow.
result Short-time existence of conformal Ricci flow on asymptotically hyperbolic manifolds.
This work considers the question of whether mean-curvature flow can be modified to avoid the formation of singularities. We analyze the finite-elements discretization and demonstrate why the original flow can result in numerical instability due to division by zero. We propose a variation on the flow that removes the nu…
The paper introduces combinatorial curvature and flow for polyhedral surfaces, proving rigidity and solving the Yamabe problem.
problem Discrete conformal structures on polyhedral surfaces and their rigidity.
method Parameterized combinatorial curvature, combinatorial α-Ricci flow, and flow extension through singularities.
result Existence and convergence of combinatorial α-Ricci flow for solving the Yamabe problem.
The paper proves Ricci flow convergence on compact manifolds with boundary.
problem Analyzing Ricci flow on compact manifolds with boundary conditions.
method Normalized Ricci flow with prescribed mean curvature on boundary.
result The solution converges to a complete hyperbolic metric with sectional curvature < -1.
Introduces conformal Bach flow and proves its well-posedness and backward uniqueness.
problem Analyzing the long-time behavior of conformal Bach flow.
method Establishes well-posedness and backward uniqueness; derives L2-estimates of curvatures. result Derives Shi's pointwise-estimate of derivatives of curvatures without assuming Sobolev constant bound.
We study the short-time existence and regularity of solutions to a boundary value problem for the Ricci-DeTurck equation on a manifold with boundary. Using this, we prove the short-time existence and uniqueness of the Ricci flow prescribing the mean curvature and conformal class of the boundary, with arbitrary initial …
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.
New metrics connect surfaces with Anosov flows to those with negative curvature.
problem Creating metrics with Anosov flows on surfaces of positive curvature.
method Constructing metrics with Anosov geodesic flows and positive curvature regions, connecting to negative curvature metrics via smooth conformal deformations.
result Existence of a smooth curve of conformal deformations connecting Anosov metrics to metrics of negative curvature.
Yamabe flow proves compactness of certain locally conformally flat manifolds with positive Ricci curvature.
problem Proving compactness of locally conformally flat manifolds with positive Ricci curvature.
method Using the Yamabe flow to prove compactness.
result Locally conformally flat manifolds with positive pinched Ricci curvature are compact.
We study a conformal flow for compact Riemannian manifolds of dimension greater than two with boundary. Convergence to a scalar-flat metric with constant mean curvature on the boundary is established in dimensions up to seven, and in any dimensions if the manifold is spin or if it satisfies a generic condition.
Study on fractional curvature flow on unit sphere, extending previous work.
problem Fractional Nirenberg problem on unit sphere.
method Fractional conformal curvature flow on unit sphere.
result Perturbation result for fractional Nirenberg problem with σ∈(1/2,1). Let Fn:(Σ,hn)→C2 be a sequence of conformally immersed Lagrangian self-shrinkers with a uniform area upper bound to the mean curvature flow, and suppose that the sequence of metrics {hn} converges smoothly to a Riemannian metric h. We show that a subsequence of {Fn} converges smoothly to …
Study negative scalar curvature metrics with positive boundary mean curvature.
problem Bounding conformal metrics with specific curvature properties.
method Analyzing Riemannian manifolds with boundary conditions.
result A priori boundedness of metrics in specific cases.
Unified flow approach to curvature problem on specific manifolds.
problem Prescribed Chern scalar curvature problem on compact Hermitian manifolds with negative Gauduchon degree.
method Unified flow approach with conditions on curvature function f. result Flow converges to a conformal Hermitian metric with specified curvature.
Study existence of conformal metrics with specific curvature properties on compact manifolds.
problem Existence of conformal metrics with constant scalar curvature and boundary mean curvature.
method Proving existence through specific cases and sequences of metrics.
result Existence of conformal metrics in various cases, including positive Yamabe constant.
In this paper we consider the Ricci flow on manifolds with boundary with appropriate control on its mean curvature and conformal class. We obtain higher order estimates for the curvature and second fundamental form near the boundary, similar to Shi's local derivative estimates. As an application, we prove a version of …
We introduce a new method to construct a large family of Lagrangian surfaces in complex Euclidean plane by means of two planar curves making use of their usual product as complex functions and integrating the Hermitian product of their position and tangent vectors. Among this family, we characterize minimal, constant m…
New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.
problem Creating new discrete conformal structures on surfaces with boundary.
method Introducing new discrete conformal structures, proving global rigidity using variational principles, and introducing combinatorial curvature flows.
result Global rigidity of new discrete conformal structures and effective algorithms for constructing hyperbolic metrics.
A well known result of Da Rios and Levi-Civita says that a closed planar curve is elastic if and only if it is stationary under the localized induction (or smoke ring) equation, where stationary means that the evolution under the localized induction equation is by rigid motions. We prove an analogous result for surface…
Variational approximations for curve flows on Riemannian manifolds.
problem Approximating solutions to curvature and elastic flow problems on Riemannian manifolds.
method Variational formulations, finite element approximations, piecewise linear elements, stability analysis.
result Derived schemes can compute rotationally symmetric self-shrinkers and geodesics.
Paper studies mean curvature flow in Minkowski spaces, proving existence and uniqueness.
problem Existence and uniqueness of mean curvature flow in Minkowski spaces.
method Introduced mean curvature flow on Finsler manifolds, proved existence and uniqueness for Minkowski spaces.
result Proved existence and uniqueness of mean curvature flow in Minkowski spaces.