The paper proves smoothness of stationary varifolds.
problem Understanding the smoothness of stationary varifolds.
method Analyzing -dimensional integer rectifiable varifolds in open sets.
result The support of stationary varifolds is rectifiable.
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48 results for “rectifiable varifold”
Rectifiable varifolds with bounded curvature can be covered by smooth surfaces.
problem Understanding the structure of rectifiable varifolds with bounded curvature.
method Using curvature of arbitrary closed sets and viscosity solutions of PDEs.
result The support of rectifiable varifolds can be covered by smooth submanifolds.
Study quantizes energy distribution in inhomogeneous phase transitions.
problem Quantifying energy distribution in inhomogeneous Allen-Cahn phase transitions.
method Analysis of varifolds and convergence of integer rectifiable varifolds.
result Equidistribution of energy between Dirichlet and Potential energy in phase field limit.
We provide a measure based topology for certain unions of C2 rectifiable submanifolds of mixed dimensions in Rn. In this topology lower dimensional sets remain in the limit as measures when higher dimensional sets collapse down to them. For example a decreasing sequence of spheres may have a limit consisting of just a …
Study the singular limit of a boundary reaction equation, showing energy concentration and varifold support.
problem Analyzing the singular limit of a boundary reaction equation.
method Investigates the critical points of the boundary reaction equation \((-Δ)^{\frac{1}{2}}u = \frac{1}{\varepsilon}(u-u^3)\) in \(U \subset \mathbb{R}^n\).
result Shows existence of an -rectifiable energy concentration set and associates limit energy measures to a stationary varifold.
The paper defines and proves the existence of decompositions of integral varifolds.
problem Existence of integral varifold decompositions.
method Introducing and proving the existence of decompositions of integral varifolds into countably many integral varifolds.
result Existence of decompositions of integral varifolds whose first variation is representable by integration.
In this work it is shown that every integral varifold in an open subset of Euclidian space of locally bounded first variation can be covered by a countable collection of submanifolds of class C^2. Moreover, the mean curvature of each member of the collection agrees with the mean curvature of the varifold almost everywh…
Study local minimizers of Ginzburg-Landau functionals in high dimensions, showing energy measures converge to rectifiable measures.
problem Investigating minimizers of Ginzburg-Landau functionals in high dimensions with energy bounds.
method Analyzing minimizers with logarithmic energy bounds and considering the vacuum manifold's homotopy classes.
result Normalized energy measures converge to an -rectifiable measure associated with a stationary varifold.
Develops a PDE approach to constructing nontrivial anisotropic surfaces.
problem Min-max construction of anisotropic surfaces.
method PDE-based approach to anisotropic surface energies.
result Construction of an anisotropic min-max hypersurface.
The paper proves inequalities for varifolds on Riemannian manifolds.
problem Proving inequalities for functions on varifolds in Riemannian manifolds.
method Developed techniques to handle functions with compact support on -rectifiable varifolds in Riemannian manifolds with positive injectivity radius and sectional curvature bounded above.
result Proved Poincaré and Sobolev type inequalities for varifolds.
Energy Concentration for Min-Max Solutions of the Ginzburg-Landau Equations on Manifolds with math.DG
Estimates Ginzburg-Landau energy to show concentration of energy on a rectifiable varifold.
problem Estimating Ginzburg-Landau energy on manifolds with nontrivial first Betti number.
method Decomposing harmonic component into integral and fractional parts, using min-max construction.
result Energy concentration on a rectifiable -varifold as .
Perpendicularity and Locality for Codimension-One Varifolds with Bounded Anisotropic Mean Curvaturemath.AP
This paper extends Euclidean theorems to anisotropic settings for varifolds.
problem Anisotropic mean curvature of codimension-one varifolds.
method Proves perpendicularity and locality of mean curvature for bounded anisotropic mean curvature varifolds.
result Anisotropic mean curvature agrees with the approximate mean curvature on the rectifiable part of the varifold.
We prove under suitable hypotheses that convergence of integral varifolds implies convergence of associated mod 2 flat chains and subsequential convergence of associated integer-multiplicity rectifiable currents. The convergence results imply restrictions on the kinds of singularities that can occur in mean curvature f…
Paper provides estimates for varifolds with critical mean curvature.
problem Estimating tilt-excess on varifolds with critical mean curvature.
method Generalizing Lipschitz approximation and Sobolev-Poincaré estimates to almost-integral rectifiable varifolds.
result VMO-type estimates for quadratic tilt-excess on varifolds with critical mean curvature.
New varifolds with capillary boundary properties studied.
problem Understanding varifolds with specific boundary conditions.
method Introducing a Radon measure on a Grassmannian bundle as a capillary boundary.
result Structural properties, monotonicity inequality, and integral compactness proved.
We develop a suitable generalization of Almgren's theory of varifolds in a lorentzian setting, focusing on area, first variation, rectifiability, compactness and closure issues. Motivated by the asymptotic behaviour of the scaled hyperbolic Ginzburg-Landau equations, and by the presence of singularities in lorentzian m…
Shows smoothness of varifolds with specific boundary angles.
problem Regularity of varifolds with prescribed contact angles.
method Analyzes varifolds with bounded first variation and prescribed contact angles, proving smoothness.
result Support of varifold is a hypersurface near the boundary.
The study examines the behavior of -harmonic maps to as approaches 2.
problem Understanding the asymptotic behavior of -harmonic maps as tends to 2.
method Analyzing the convergence of singular sets and stationary varifolds.
result The singular sets of -harmonic maps converge to a stationary rectifiable varifold of density .
Graphs with bounded anisotropic mean curvature are regular almost everywhere.
problem Understanding the regularity of graphs with anisotropic mean curvature.
method Proving regularity for -dimensional Lipschitz graphs with anisotropic mean curvature bounded in .
result Graphs with bounded anisotropic mean curvature are regular almost everywhere.
The present paper is intended to provide the basis for the study of weakly differentiable functions on rectifiable varifolds with locally bounded first variation. The concept proposed here is defined by means of integration by parts identities for certain compositions with smooth functions. In this class the idea of ze…
This paper introduces first order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally nonlinear class of generalised weakly differentiable functions and share key functional analytic properties with their Euclidean counterparts. Assuming the varifold to s…
We prove existence and partial regularity of integral rectifiable -dimensional varifolds minimizing functionals of the type and in a given Riemannian -dimensional manifold , and , under suitable assumptions on (in the end of the paper we give many examples of …
The paper studies surfaces in a bounded domain with orthogonal boundaries and proves curvature estimates.
problem Estimating the area of surfaces with orthogonal boundaries in a bounded domain.
method Weak formulation of orthogonality for curvature varifolds, classification of vanishing curvature varifolds.
result Existence of an orthogonal 2-varifold that minimizes curvature in the integer rectifiable class.
The Structure of Stable Codimension One Integral Varifolds near Classical Cones of Density 5/2math.DG
Proves regularity for stable varifolds near specific cones.
problem Regularity of stable codimension one integral varifolds near certain cones.
method Develops blow-up arguments and inductively performs finer blow-up procedures.
result Proves regularity for varifolds close to specific cones.
If one considers an integral varifold with bounded mean curvature, and if $S^k(I)\equiv\{x\in M: \text{ no tangent cone at $x$ is }k+1\text{-symmetric}\}$ is the standard stratification of the singular set, then it is well known that . In complete generality nothing else is known about …
We find nontrivial solutions to a Ginzburg-Landau equation on compact manifolds.
problem Finding nontrivial solutions to a specific Ginzburg-Landau equation on compact manifolds.
method Using min-max techniques to construct solutions whose energy grows logarithmically with a small parameter.
result The energy of constructed solutions concentrates on a nontrivial stationary, rectifiable -varifold.
Develops weak formulation for spacelike flows in pseudo-Euclidean space.
problem Weak formulation of spacelike mean curvature flow in pseudo-Euclidean space.
method Based on spacelike integer rectifiable varifolds and pseudo-Euclidean first variation.
result Existence and compactness of spacelike Brakke flows with fixed boundary.
Approximate mean curvature flows of a general varifold, and their limit spacetime Brakke flowmath.DG
Constructs approximate mean curvature flows for general varifolds.
problem Mean curvature flow for general initial data.
method Approximation of mean curvature flows using varifolds and iterated push-forwards.
result Approximate mean curvature flow converges to a spacetime Brakke flow under certain conditions.
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.
Connected surfaces with boundary minimize Willmore energy under certain conditions.
problem Finding connected compact surfaces with minimal Willmore energy.
method Minimizing the Willmore energy on integer rectifiable curvature varifolds with boundary constraints.
result Existence of connected minimizers when the infimum of the problem is less than 4π.
The paper defines new ways to measure higher-order differentiability of sets.
problem Measuring differentiability of arbitrary sets in Euclidean space.
method Develops two concepts of pointwise differentiability using distance functions to smooth submanifolds.
result Strong pointwise differentiability of every positive integer order at almost all points of the set's intersection with a plane.
A mean value inequality for the generalized self-expander type submanifolds and its applicationmath.DG
In this paper we get a version of mean value inequality for generalized self-expander type submanifolds in Euclidean space. As the application, we prove that if mean curvature flow on the self-expander in Euclidean space subconverges to an -rectifiable varifold in weak sense for goes to the singular t…
Study shows non-uniqueness of Brakke flow near flat singular points.
problem Exploring instability of minimal surfaces at flat singular points.
method Analyzes the behavior of stationary varifolds and their blow-ups.
result Proves existence of non-constant Brakke flow near flat singular points.
Brakke flow support is parabolically rectifiable
problem Support of Brakke flow is parabolically rectifiable
method Developed approach to Brakke flow as space-time-Grassmann measure
result Standard convergence of Brakke flows is equivalent to space-time-Grassmann Radon measures
Michael-Simon inequality proven for anisotropic energies close to area.
problem Proving Michael-Simon inequality for anisotropic integrands close to area.
method New functional inequality for vector fields on the plane, simplifying Almgren's proof.
result Michael-Simon inequality holds for convex anisotropic integrands close to 1.
New theory for area of Legendrian surfaces, proving smoothness and variational results.
problem Understanding the area of Legendrian surfaces under constraints.
method Introducing PHSLVs, proving sequential compactness, regularity, and variational results.
result Generalized regularity theory for Legendrian surfaces, achieving variational minima.
In this paper, we address the problem of orientation that naturally arises when representing shapes like curves or surfaces as currents. In the field of computational anatomy, the framework of currents has indeed proved very efficient to model a wide variety of shapes. However, in such approaches, orientation of shapes…
Study of Ginzburg-Landau equation on manifolds with boundary, showing energy breakdown.
problem Behavior of solutions to Ginzburg-Landau equation on manifolds with boundary.
method Asymptotic analysis, energy upper bound, convexity condition, harmonic 1-form, rectifiable (n-2)-varifold.
result Energy of solutions breaks into two parts: one captured by a harmonic 1-form and the other by a stationary rectifiable (n-2)-varifold.
The study examines stationary integral varifolds near multiplicity 2 planes, proving regularity under specific conditions.
problem Understanding the structure of stationary integral varifolds near multiplicity 2 planes.
method Investigates the structure of varifolds close to planes with multiplicity 2, proving an ε-regularity theorem under certain conditions.
result In , is represented by the graph of a Lipschitz 2-valued function over with small Lipschitz constant; all tangent cones at singular points are unique and comprised of stationary unions of 4 half-planes.
The paper establishes structure theory for stable varifolds with applications to area minimising hypersurfaces.
problem Understanding the structure of stable codimension 1 integral varifolds.
method Develops a structure theory for stable codimension 1 stationary integral varifolds with no classical singularities.
result Establishes local structure properties of area minimising currents mod p, including the uniqueness of tangent cones at points with planar tangent cones.
For Ginzburg-Landau vortices, energy quantization holds only when density is less than 2.
problem Energy quantization in Ginzburg-Landau vortices for higher dimensions.
method Analyzing normalized energy measures and vorticity sets.
result Energy quantization only holds when density is less than 2.
A Curvature Varifold Whose Weak Second Fundamental Form is Not Preserved Under Decompositionsmath.DG
New varifold example shows decomposition failure.
problem Curvature varifolds cannot always be decomposed.
method Constructed a specific curvature varifold.
result Found a varifold with a non-preserved weak second fundamental form under decomposition.
The concept of varifoldmath.DG
Survey of varifolds with focus on their regularity.
problem Understanding the concept and regularity of varifolds.
method Survey and examples of varifolds, emphasizing mean curvature and integral varifolds.
result Emphasis on the regularity of integral varifolds with mean curvature.
We show that metrics that maximize the k-th Steklov eigenvalue on surfaces with boundary arise from free boundary minimal surfaces in the unit ball. We prove several properties of the volumes of these minimal submanifolds. For free boundary minimal submanifolds in the ball we show that the boundary volume is reduced up…
New moving plane method for varifolds promotes smoothness from boundary to interior.
problem Promoting smoothness from boundary to interior for singular hypersurfaces.
method Introduced a moving plane method for varifolds, showing smoothness as a conclusion.
result Smoothness and symmetry in the interior can be promoted from smoothness and symmetry at infinity.
Upper bound for Morse index of min-max varifolds.
problem Bounding Morse index of varifolds.
method Proving upper bound for Morse index of min-max stationary integral varifolds.
result Upper bound for Morse index of min-max stationary integral varifolds.
This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space with its first variation given by either a Radon measure or a function in some Lebesgue space. Pointwise decay results for the quadratic tilt-excess are established for those varifolds. The results are optimal in terms of…
The paper simplifies arguments for stationary varifolds results.
problem Height bound and Lipschitz approximation for stationary varifolds.
method Simpler arguments to obtain height bound and Lipschitz approximation.
result Excess decay as a consequence of height bound and Lipschitz approximation.