Minimal surfaces' boundary points are always smooth.
problem Boundary regularity of minimal surfaces.
method Proving all boundary points are regular submanifolds.
result Boundary points of minimal surfaces are regular.
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48 results for “Boundary regularity”
Allard-type regularity theory for area minimizing currents at boundaries with arbitrary multiplicitymath.AP
Study improves boundary smoothness for area-minimizing currents with complex boundaries.
problem Boundary regularity for area-minimizing currents with arbitrary multiplicity.
method Generalization of Allard's boundary regularity theorem.
result Derivation of structural consequences from the generalized theorem.
The study shows that certain graphs are regular at boundary points.
problem Boundary regularity of anisotropic minimal Lipschitz graphs.
method Proves regularity for graphs with bounded anisotropic mean curvature and atomic energy condition.
result Regularity at boundary points with density bounded above by 1/2 + σ.
Proves smoothness of minimal surfaces near polyhedral boundaries.
problem Smoothness of free-boundary minimal surfaces near polyhedral domains.
method Allard-type regularity theorem for minimal surfaces in convex polyhedra.
result Minimal surfaces are graphical over a free-boundary plane if close to it.
Study optimal transport on simplex boundary, proving transport map and potential regularity.
problem Regularity of transport map and potential on simplex boundary.
method Boundary regularity results for optimal transport maps, exploiting simplex symmetries.
result Regularity properties of transport map and its convex potential.
Classifies local boundary conditions for Dirac-type operators on manifolds.
problem Determining all local smooth boundary conditions for Dirac-type operators.
method Combining general theory of boundary value problems for Dirac operators and pointwise considerations.
result Classification of local self-adjoint regular boundary conditions for Dirac spinors in dimensions 3 and 4.
We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; t…
We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; t…
Study on regularity of optimal transport maps on convex domains with quadratic cost.
problem Regularity of optimal transport maps between convex domains with quadratic cost.
method Analysis of -densities and boundary conditions, monotonicity formula for optimal transport maps.
result Proves -regularity for nondegenerate -densities and -regularity for boundary.
We study boundary regularity for conformally compact Einstein metrics in even dimensions by generalizing the ideas of Michael Anderson. Our method of approach is to view the vanishing of the Ambient Obstruction tensor as an nth order system of equations for the components of a compactification of the given metric. This…
We prove regularity results up to the boundary for time independent generalized Maxwell equations on Riemannian manifolds with boundary using the calculus of alternating differential forms. We discuss homogeneous and inhomogeneous boundary data and show 'polynomially weighted' regularity in exterior domains as well.
Uniform Shapiro-Lopatinski conditions and boundary value problems on manifolds with bounded geometrymath.AP
We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem is equivalent to the uniform regularity of the natural family of associated boundary value …
Study shows how maps from certain geometric spaces behave near their edges.
problem Boundary regularity of harmonic maps in specific geometric spaces.
method Analysis of and spaces.
result Established boundary regularity of harmonic maps.
New static vacuum metrics confirmed for near Euclidean boundary data.
problem Establishing sufficient conditions for near Euclidean boundary data in static vacuum metrics.
method Using new arguments from studying the conjecture for arbitrary static vacuum metrics.
result Any hypersurface in a dense subfamily is static regular.
We establish the regularity theory for certain critical elliptic systems with an anti-symmetric structure under inhomogeneous Neumann and Dirichlet boundary constraints. As applications, we prove full regularity and smooth estimates at the free boundary for weakly Dirac-harmonic maps from spin Riemann surfaces. Our met…
Proves ε-regularity for capillary surfaces in Riemannian manifolds.
problem Regularity of minimal surfaces with capillary boundary conditions.
method Uniform first variation control and ε-regularity theorems for varifolds.
result Capillary varifolds with bounded mean curvature and close to a capillary half-plane coincide with a properly embedded hypersurface.
A 3D area-minimizing current in R^5 has a 2-fold essential singularity.
problem Understanding singularities in area-minimizing currents.
method Construction of a specific current with prescribed boundary properties.
result The boundary regularity theory is dimensionally sharp.
This paper improves boundary regularity of harmonic maps in metric measure spaces.
problem Improving boundary regularity of harmonic maps in non-smooth spaces.
method Developed a Gauss-Green formula for spaces and applied it to harmonic maps.
result Optimal boundary regularity of harmonic maps from -spaces to -spaces.
Sharp generalization of boundary regularity for area minimizing currents with arbitrary multiplicity.
problem Boundary regularity of area minimizing currents with multiplicity.
method Sharp generalization of Allard's boundary regularity theorem to higher multiplicity settings.
result The set of density singular boundary points of is -rectifiable.
Optimal $C^{1, rac{1}{2}}$-regularity for -surfaces with free boundary.
problem Proving optimal regularity for -surfaces with free boundary.
method Analyzing -surfaces with free boundary on a -manifold.
result Optimal $C^{1, rac{1}{2}}$ regularity up to the boundary.
We develop a theory of surfaces with boundary moving by mean curvature flow. In particular, we prove a general existence theorem by elliptic regularization, and we prove boundary regularity at all positive times under very mild hypotheses.
Optimally regularizes boundaries in the Heisenberg group with prescribed curvature.
problem Optimizing boundaries with prescribed sub-Finsler mean curvature in the Heisenberg group.
method Analyzes critical sets of the prescribed mean curvature functional in the Heisenberg group.
result Characteristic curves of critical sets are -regular, optimal in the Heisenberg group.
The paper proves smoothness of almost-minimizers' boundaries near the free boundary.
problem Minimizing degenerate area functionals with weighted boundary conditions.
method Epsilon-regularity theorem applied to almost-minimizers.
result Almost-minimizers' boundaries are -smooth, orthogonal to the boundary .
We develop the notion of Brakke flow with free-boundary in a barrier surface. Unlike the classical free-boundary mean curvature flow, the free-boundary Brakke flow must "pop" upon tangential contact with the barrier. We prove a compactness theorem for free-boundary Brakke flows, define a Gaussian monotonicity formula v…
Study Dirac operators on finite warped cylinders with gauge fields.
problem Characterize spectral flow on finite warped cylinders with gauge fields.
method Identify endpoint operators, derive determinant characterization, introduce regularized APS conditions.
result Regularized APS conditions admit a spectral-flow framework, matching zero-mode sets.
Study shows unique tangent cones for area-minimizing currents at boundary points.
problem Uniqueness of tangent cones for area-minimizing currents with arbitrary multiplicity.
method Analysis of area minimizing currents in submanifolds with arbitrary boundary multiplicity.
result Tangent cones are unique at density boundary points.
On compact 2-manifolds with non-empty convex boundary, we prove a regularity result for integral 1-varifolds that are stationary with free boundary and -almost minimizing in small annuli. That regularity says that is a free boundary finite geodesic network. Next, using that regularity, we compute …
Study on moduli spaces of Seiberg-Witten equations on manifolds with boundary.
problem Analyzing moduli spaces of Seiberg-Witten equations on manifolds with boundary.
method General regularity theorem, strong unique continuation principle, and gluing theorem for Dirac operators; smoothness of restriction map.
result Proves moduli spaces are Hilbert manifolds and have semi-infinite-dimensionality properties.
We introduce a notion of non-local almost minimal boundaries similar to that introduced by Almgren in geometric measure theory. Extending methods developed recently for non-local minimal surfaces we prove that flat non-local almost minimal boundaries are smooth. This can be viewed as a non-local version of the Almgren-…
New PL-invariants defined for 4-manifolds with boundary.
problem Understanding the structure of PL 4-manifolds with boundary.
method Introducing weighted regular genus and weighted G-degree, proving inequalities, computing bounds.
result Lower bounds for weighted G-degree and weighted regular genus are improved.
Study anisotropic obstacle problem for minimal surfaces using Cahn-Hoffman transform.
problem Anisotropic obstacle problem for minimal surfaces.
method Cahn-Hoffman transform to convert to isotropic problem with generalized Robin boundary condition.
result Optimal regularity of the solution and regularity of the free boundary.
Study on high-codimensional minimal surfaces in hyperbolic space.
problem Understanding high-codimensional minimal surfaces in hyperbolic space.
method Investigating asymptotic behavior and boundary regularity of area-minimizing currents.
result Established boundary regularity results for high-codimensional minimal surfaces near their asymptotic boundaries.
Study boundary behavior of limit interfaces in Riemannian manifolds without convexity assumptions.
problem Boundary behavior of limit interfaces in Riemannian manifolds.
method Proves limit-interface is a free boundary varifold, integer rectifiable up to boundary.
result No convexity assumption required; valid even when limit-interface clusters near boundary.
Study on free boundary problems in RCD spaces, proving existence and regularity.
problem Free boundary problems in RCD metric measure spaces.
method Existence and local Lipschitz regularity of solutions, free boundary analysis.
result Existence and regularity of solutions, free boundary structure.
Study submanifolds with boundaries in Heisenberg groups using Stokes' Theorem.
problem Understanding submanifolds with boundaries in Heisenberg groups.
method Introduce and study -regular submanifolds with boundaries, prove Stokes' Theorem for them.
result Prove a version of Stokes' Theorem for submanifolds with boundaries in Heisenberg groups.
We study various aspects related to boundary regularity of complete properly embedded Willmore surfaces in H3, particularly those related to assumptions on boundedness or smallness of a certain weighted version of the Willmore energy. We prove, in particular, that small energy controls C1 boundary regularity. We examin…
Lower bounds for PL 4-manifolds with boundary are improved.
problem Estimating PL 4-manifolds with boundary.
method Proved inequalities for regular genus and gem-complexity.
result Improved lower bounds for PL 4-manifolds with boundary.
In this paper we establish two boundary versions of the Schwarz lemma. The first is for general holomorphic self maps of bounded convex domains with boundary. This appears to be the first boundary Schwarz lemma for general holomorphic self maps that requires no strong pseudoconvexity or finite type assumptions. T…
In this paper, we study the regularity of asymptotically hyperbolic metrics with Einstein condition near boundary and Weyl curvature smooth enough in arbitrary dimension. Following Michael Anderson's method, we show that conformally compact Riemannian metrics with Einstein equation vanishing to finite order n…
Study of elliptic boundary value problems on non-compact manifolds.
problem Analyzing elliptic differential operators on manifolds with non-compact boundaries.
method Regularity theory and trace theorems for sections in the maximal domain under various assumptions.
result Systematic study of local and nonlocal boundary conditions, including the Atiyah-Patodi-Singer condition.
Boundary Estimates for the Monge-Ampère Equation in the Polygons with Guillemin Boundary Conditionsmath.AP
Extends boundary estimates for Monge-Ampère equations in polygonal domains.
problem Boundary regularity for Monge-Ampère equations on convex polytopes with specific boundary conditions.
method Schauder-type techniques, inspired by Donaldson's work on the Abreu equation.
result Establishes boundary regularity result for Hölder continuous right-hand sides.
r-Regularitymath.DG
We provide a characterization of r-regular sets in terms of the Lipschitz regularity of normal vector fields to the boundary.
Study heat flow for half-harmonic maps and harmonic maps with free boundary.
problem Integrability and regularity of half-harmonic maps and harmonic maps with free boundary.
method Introduced a heat flow associated to half-harmonic maps and constructed weak solutions via Ginzburg-Landau approximation.
result Proved partial regularity of weak solutions in space and time.
Proves multiplicity one for boundary minimal hypersurfaces in compact manifolds.
problem Proving multiplicity one for min-max free boundary minimal hypersurfaces in compact manifolds with boundary.
method Developed existence and regularity theory for free boundary hypersurfaces with prescribed mean curvature, including Morse index bounds.
result Proved multiplicity one theorem for min-max free boundary minimal hypersurfaces in compact manifolds with boundary.
Solves Einstein constraint equations on compact manifolds with specified boundaries.
problem Solving Einstein constraint equations with specified boundaries.
method Studies conformal constraint equations with low regularity assumptions.
result Solves Einstein constraint equations on compact manifolds with specified boundaries.
Unique solutions found for Plateau problems in smooth and continuous calibrations.
problem Finding unique solutions to the Plateau problem for specific types of currents.
method Boundary regularity theory for area-minimizing currents and unique continuation argument.
result Every compactly supported smoothly or continuously calibrated integral current is the unique solution to the Plateau problem for its boundary data.
The paper examines the structure and stability of boundaries in noncollapsed RCD spaces.
problem Structuring and stability of boundaries in noncollapsed RCD spaces.
method Effective measure bounds and ε-regularity theorem.
result The boundary is homeomorphic to a manifold away from a set of codimension 2 and is rectifiable.
Study submanifolds in hyperbolic space, focusing on their boundary and Laplace operator.
problem Understanding the geometry and regularity of submanifolds in hyperbolic space.
method Analyzing asymptotic geometry and regularity properties near the ideal boundary, computing essential spectra.
result Computed essential spectra of the Laplace operator on certain submanifolds.