Simply connected surfaces with large constant mean curvature and free boundaries concentrate at critical points of the boundary's mean curvature.
problem Surfaces with large constant mean curvature and free boundaries.
method Proving concentration at critical points of the boundary's mean curvature.
result Simply connected H-surfaces concentrate at critical points of the boundary's mean curvature.
Study constant mean curvature surfaces with integrable boundary conditions.
problem Understanding surfaces with constant mean curvature under specific boundary conditions.
method Used generalized Weierstrass representation to determine potentials.
result Determined potentials for surfaces satisfying integrable boundary conditions.
Open problems on surfaces with boundary and constant mean curvature.
problem Open problems on compact constant mean curvature surfaces with boundary.
method Survey of current status of open problems.
result Collection of open problems in the theory of surfaces.
Study of prescribing scalar curvature and mean curvature on compact manifolds with boundary.
problem Prescribing scalar curvature and mean curvature on compact manifolds with boundary.
method Introducing singular metrics inspired by previous work on closed manifolds, proving rigidity results for flat manifolds with totally geodesic boundary.
result Generic scalar-flat manifolds with minimal boundary can have scalar curvature and mean curvature prescribed simultaneously.
The paper proves the existence of constant mean curvature disks with capillary boundary conditions.
problem Existence of constant mean curvature disks with specific boundary conditions.
method Extending Struwe's result to a broader range of boundary angles.
result Existence of constant mean curvature disks with index at most 1.
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.
The paper examines stability of Yamabe boundary problem under perturbations.
problem Stability of Yamabe boundary problem under perturbations of mean curvature and scalar curvature.
method Analyzes stability of the Yamabe boundary problem with respect to perturbations of mean curvature and scalar curvature.
result The stability of the Yamabe boundary problem is proven under perturbations from below, but not from above.
Geodesic boundaries of surfaces with constant mean curvature are studied.
problem Understanding geodesic boundaries of surfaces with constant mean curvature.
method Generalization of results from minimal surfaces to constant mean curvature surfaces.
result Geodesic boundaries of constant mean curvature surfaces are explored.
The paper studies curvature changes on manifolds with boundary.
problem Investigating conformal deformations of curvature on manifolds with boundary.
method Establishing sufficient conditions for positive scalar curvature and mean convex boundary, exploring further deformation scenarios.
result Conditions for conformal deformations to complete metrics with positive scalar curvature and mean convex boundary.
Proves rigidity of boundaries with constant mean curvature in warped product manifolds.
problem Rigidity and compactness of boundaries with constant mean curvature in warped product manifolds.
method Distributional CMC-rigidity proof for rectifiable boundaries.
result Characterizes limits of boundaries with converging mean curvatures.
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.
Study ancient solutions to free boundary mean curvature flow in convex manifolds.
problem Understanding ancient solutions to free boundary mean curvature flow in convex manifolds.
method Establish rigidity results and construct foliations to describe ancient solutions.
result Ancient solutions to free boundary mean curvature flow are rigid and exhaust all possibilities under certain conditions.
Compact metrics found with specific curvature properties on 3D surfaces.
problem Finding compact metrics with constant curvature on 3D surfaces.
method Blow-up analysis of Yamabe equation with critical Sobolev exponents.
result Proved the compactness of conformal metrics with constant scalar curvature and boundary mean curvature.
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…
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 study topological obstructions to the existence of a Riemannian metric on manifolds with boundary such that the scalar curvature is non-negative and the boundary is mean convex. We construct many compact manifolds with boundary which admit no Riemannian metric with non-negative scalar curvature and mean convex bound…
Inverse mean curvature flow converges to a disk in hyperbolic space.
problem Understanding flow behavior in hyperbolic geometry.
method Inverse mean curvature flow with free boundary on geodesic spheres.
result Flow converges to a totally geodesic disk.
Proves existence of mean curvature flow with surgery for free boundary surfaces.
problem Existence of mean curvature flow with free boundary.
method New approach for flows with surgery in free boundary setting.
result Flow converges to stable minimal surfaces without surgeries for large t.
Study on compactness and blow-up of solutions for Yamabe problems on manifolds with non-umbilic boundaries.
problem Compactness and blow-up behavior of solutions to the Yamabe boundary problem on manifolds with non-umbilic boundaries.
method Analysis of stability and blow-up sequences for solutions under perturbations of mean curvature and scalar curvature.
result Existence of a blowing-up sequence of solutions when perturbing the mean curvature from above or below with a function having a large positive maximum.
Unique ancient convex flow in a ball with free boundary found.
problem Classifying convex ancient free boundary mean curvature flows in the ball.
method Proof of existence and uniqueness in every dimension.
result A unique (modulo rotations and translations) convex ancient mean curvature flow found.
Study finds conditions for metrics on curved spaces.
problem Finding metrics with specific curvature properties.
method Analyzes a conformal class with prescribed scalar and boundary mean curvatures.
result Establishes a necessary and sufficient condition involving a conformal invariant.
Constructs obstructions and deformation principles for positive scalar curvature metrics with mean convex boundaries.
problem Obstructing the existence of positive scalar curvature metrics with mean convex boundaries.
method Atiyah-Patodi-Singer index formula, deformation principle, homotopy equivalences, higher homotopy groups.
result Construction of compact manifolds with nontrivial higher homotopy groups for positive scalar curvature metrics with mean convex boundaries.
Localized deformation of scalar curvature and mean curvature on manifolds.
problem Deforming scalar curvature and mean curvature on compact manifolds with boundary.
method Proving localized surjection of scalar curvature and mean curvature map, handling non-variational linearized problem.
result Localized deformations of scalar curvature and mean curvature on compact manifolds are possible.
Paper finds unique solutions for curved surfaces with specific gradient.
problem Existence of curved surfaces with specific gradient.
method Second boundary value problem of constant mean curvature equations.
result Unique convex solutions for constant mean curvature equations.
Classifies metrics with specific curvature properties on a ball.
problem Classifying conformal metrics with constant σk curvature and constant boundary mean curvature. method Uses the Obata-Escobar argument to classify metrics on the upper hemisphere.
result Extends a result of Escobar for k=1 to include positive and negative cones. Unbounded convex domains have zero mean curvature on disconnected boundaries.
problem Understanding mean curvature in unbounded convex domains.
method Analyzing mean curvature on disconnected boundary components.
result Mean curvature is zero on disconnected boundary components of unbounded mean convex domains.
We discuss some consequences of the existence of the holomorphic quadratic Hopf differential on a conformally immersed constant mean curvature topological disc with analytic boundary. In particular, we derive a formula for the mean curvature as a weighted average of the normal curvature of the boundary curve, and a con…
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.
Develops slicing method to prove rigidity of scalar curvature on manifolds with boundary.
problem Understanding positive scalar curvature metrics on manifolds with boundary.
method Minimal slicing via capillary hypersurfaces to prove rigidity statements.
result Proves rigidity statement in dimension 4 for specific geometric conditions.
The study proves a rigidity theorem for compact manifolds with boundary.
problem Rigidity of compact manifolds with boundary in low dimensions.
method Dimension reduction argument for mean curvature, extending Schoen-Yau's for scalar curvature.
result Sharp spherical radius rigidity and best NNSC fill-in in terms of mean curvature.
We prove the convexity estimates of Huisken-Sinestrari for finite-time singularities of mean-convex, mean curvature flow with free boundary in a barrier S. Here S can be any properly embedded, oriented surface in Rn+1 of bounded geometry. We also give an alternative proof that convex mean curvature flows with …
Study on static manifolds with boundary and rigidity of curvature.
problem Understanding the rigidity of scalar curvature and mean curvature on manifolds with boundary.
method Analyzing maps of scalar curvature in the interior and mean curvature on the boundary, discussing geometric properties of static manifolds.
result Classification and rigidity theorems for simple non-generic domains in space forms and Schwarzschild manifold.
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 proves existence and nonexistence for annular surfaces with specific curvature and boundary.
problem Existence and nonexistence of annular surfaces with prescribed mean curvature.
method Proves existence and nonexistence results for normal graphs of unduloids or nodoids.
result Existence and nonexistence results for annular type surfaces with prescribed mean curvature.
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.
New theorem on 3-manifolds with curvature and convex boundary.
problem Understanding 3-manifolds with specific curvature and boundary properties.
method Analyzes properties of Riemannian 3-manifolds with nonnegative scalar curvature and mean-convex boundary.
result Shows flatness of certain 3-manifolds containing specific geometric objects.
We establish a gluing theorem for solutions of a Yamabe problem for manifolds with boundary studied by Escobar in the 90's. Given two scalar-flat Riemannian manifolds whose boundary has zero mean curvature and sharing a submanifold K, we produce the generalized connected sum along K. On this third manifold we produ…
The study classifies and constructs examples of surfaces with specific curvature and boundary conditions.
problem Classifying surfaces with parallel mean curvature and constant contact angle.
method Analytical and geometric methods, including classification and construction of examples.
result Sharp classification and examples of branched immersed disks and surfaces in space forms.
Develops a method to deform metrics on manifolds with non-compact boundaries.
problem Creating metrics with positive scalar curvature on manifolds with boundary.
method General deformation principle for Riemannian metrics on manifolds with non-compact boundaries.
result Non-existence of metrics with positive scalar curvature and mean convex boundary.
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…
In this paper we introduce a Guan-Li type volume preserving mean curvature flow for free boundary hypersurfaces in a ball. We give a concept of star-shaped free boundary hypersurfaces in a ball and show that the Guan-Li type mean curvature flow has long time existence and converges to a free boundary spherical cap, pro…
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.
Study of star-shaped hypersurfaces with capillary boundary using constrained mean curvature flow.
problem Understanding the evolution of hypersurfaces with capillary boundaries.
method Locally constrained mean curvature flow for star-shaped hypersurfaces in the half-space.
result Established new Alexandrov-Fenchel inequalities for convex hypersurfaces with capillary boundary.
The purpose of this paper is twofold: firstly, to establish sufficient conditions under which the mean curvature flow supported on a hypersphere with exterior Dirichlet boundary exists globally in time and converges to a minimal surface, and secondly, to illustrate the application of Killing vector fields in the preser…
Study on well-posedness of vacuum Einstein equations with specific boundary conditions.
problem Well-posedness of the initial boundary value problem for vacuum Einstein equations with geometric boundary conditions.
method Analysis of conformal-mean curvature boundary data, proving dense solution space and Holmgren-type uniqueness theorem.
result Linearized problem has a solution space with dense range in C∞, valid for general smooth linearized solutions. Study finds solutions to curvature equation with boundary conditions.
problem Finding solutions to curvature equation with boundary conditions.
method Established local C2 estimates and used blow-up analysis. result Existence of conformal metrics with prescribed curvature and boundary conditions.
The paper studies singularities in mean curvature flow and bounds on minimal surface total curvature.
problem Understanding singularities in mean curvature flow and bounds on minimal surface total curvature.
method Mean curvature flow and geometric analysis.
result The largest number k for which a minimal surface with total curvature less than k is a disk is greater than 3π.
Paper proves gradient estimates for Lagrangian mean curvature equation.
problem Proving gradient estimates for Lagrangian mean curvature equation.
method Interior gradient estimates for critical and supercritical Lagrangian mean curvature equation.
result Solves Dirichlet boundary value problem for critical and supercritical Lagrangian mean curvature equation.