Green functions on stationary varifolds established with inequalities and convergence results.
problem Establishing Green functions on stationary varifolds with inequalities and convergence.
method Extending Grüter and Widman's method, constructing Green functions, using local Harnack inequality.
result Green functions converge for sequences of stationary varifolds converging with multiplicity one.
Paper shows convergence types match for almost minimal sets.
problem Matching convergence types for almost minimal sets.
method Hausdorff and varifold convergence comparison on almost minimal sets.
result Hausdorff and varifold convergence coincide on almost minimal sets.
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 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…
New varifold solutions for mean curvature flow converge and are unique.
problem Mean curvature flow and Allen-Cahn equation convergence and uniqueness.
method Evolving varifolds coupled to phase volumes, weak-strong uniqueness principle.
result Limits of Allen-Cahn solutions are varifold solutions, and classical flows are unique.
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.
This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic notions which have been applied to study sequences of submanifolds in Euclidean…
The study introduces flows for lagrangian varifolds to converge to special lagrangian cycles.
problem Understanding the behavior of lagrangian cycles in Calabi-Yau manifolds.
method Constructing mass-decreasing flows of lagrangian varifolds and cycles with vanishing Maslov index.
result Special lagrangian cycles are generated by the flow of cycles, and they represent a part of the lagrangian homology.
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.
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 (n−1)-rectifiable energy concentration set and associates limit energy measures to a stationary varifold. 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 (n−2)-rectifiable measure associated with a stationary varifold. Compactness results for hypersurfaces with mean curvature prescribed by an ambient function.
problem Proving compactness for hypersurfaces with prescribed mean curvature.
method Using oriented integral varifolds and a weak notion of curvature coefficients.
result Locally uniform bounds on second fundamental form lead to compactness.
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.
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 study bounds the topology of free boundary minimal surfaces in 3D manifolds.
problem Understanding the topology of free boundary minimal surfaces in compact 3D manifolds.
method Establishing general bounds on the topology via min-max methods and analyzing varifolds.
result The first Betti number is lower semicontinuous in the limit of min-max sequences.
The paper studies minimal submanifolds from the abelian Higgs model, proving convergence of energy measures and currents.
problem Existence and properties of minimal submanifolds from the abelian Higgs model.
method Analyzes rescalings of the self-dual Yang-Mills-Higgs energy, showing convergence of energy measures and currents.
result Provides a variational construction of nontrivial critical points and proves the existence of stationary integral (n-2)-varifolds.
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.
The study examines the behavior of p-harmonic maps to S1 as p approaches 2.
problem Understanding the asymptotic behavior of p-harmonic maps as p tends to 2. method Analyzing the convergence of singular sets and stationary varifolds.
result The singular sets of p-harmonic maps converge to a stationary rectifiable varifold of density 2π. 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.
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.
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.
Generalizes Reilly inequality to varifolds and analyzes equality cases.
problem Extending Reilly inequality to varifolds.
method Generalization of Reilly inequality to H(2) varifolds and polygons. result Analyzed the equality cases of the generalized inequality.
Counterexample and new proof for curvature varifolds.
problem Counterexample to Hutchinson's proof and new proof of C1,α representation. method Alternative proof method and decomposition of varifolds.
result Structure theorem for curvature varifolds with null second fundamental form.
Sharp criteria for 2-varifolds to be induced by smooth immersions.
problem Regularity of integral 2-varifolds with square integrable mean curvature.
method Fine analysis of Hausdorff density and recent local regularity results.
result Optimal threshold for Willmore energy leading to curvature varifolds.
Establishes a boundary maximum principle for varifolds with fixed contact angle.
problem Boundary behavior of varifolds with contact angle constraints.
method Maximum principle for stationary pairs of varifolds with fixed contact angle condition.
result Boundary maximum principle proven for stationary varifolds.
Extends Campanato theory to multi-valued functions for geometric variational problems.
problem Regularity of multi-valued functions in geometric variational problems.
method Adapting Campanato's ideas to multi-valued functions, proving regularity theorems.
result Established regularity for multi-valued harmonic functions and stationary integral varifolds.
Sharp lower bound found for integral varifolds' mean curvature.
problem Finding a sharp lower bound for the mean curvature integral of integral varifolds.
method Developed a new approach using integral varifolds and mean curvature.
result A sharp lower bound on the mean curvature integral with critical power for integral varifolds.
Alternative proof of weak solutions to mean curvature flow using minimizing movements.
problem Existence of weak solutions to mean curvature flow and volume preserving mean curvature flow.
method Proposes a new existence proof using a minimizing movements scheme and a novel proxy for distance.
result Unconditional convergence towards a De Giorgi solution for the minimizing movements scheme.
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.
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…
The paper studies the consistency of mean curvature flow via volumetric varifolds.
problem Consistency of mean curvature flow.
method Discretization using volumetric varifolds and derivation of Brakke approximate equality.
result Derivation of a Brakke approximate equality involving varifold masses and approximate mean curvatures.
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 C1,α properly embedded hypersurface. Stationary polyhedral varifolds minimize area in two senses.
problem Minimizing area of polyhedral varifolds.
method Proves minimization of area through specific conditions.
result Stationary polyhedral varifolds minimize area in two senses.
Optimal regularity for 2D stationary varifolds proved.
problem Regularity of parametrized integer stationary varifolds in 2D.
method Established optimal regularity result for stationary varifolds.
result Smooth minimal branched immersion and constant multiplicity function.
Study introduces indecomposability for varifolds, leading to geometric consequences.
problem Understanding the structure of varifolds and their connectedness properties.
method Introducing indecomposability and related concepts for varifolds.
result Substantial geometric consequences derived from the connectedness properties of varifolds.
We present the min-max construction of critical points of the area using penalization arguments. Precisely, for any immersion of a closed surface Σ into a given closed manifold, we add to the area Lagrangian a term equal to the Lq norm of the second fundamental form of the immersion times a "viscosity" parameter. …
Paper shows how solutions to Allen-Cahn converge to multiphase mean curvature flow.
problem Convergence of Allen-Cahn solutions to multiphase mean curvature flow.
method Conditional convergence result of Allen-Cahn solutions to De Giorgi type BV-solutions of multiphase mean curvature flow.
result De Giorgi type BV-solutions are unique in a weak-strong sense.
The paper proves smoothness of stationary varifolds.
problem Understanding the smoothness of stationary varifolds.
method Analyzing m-dimensional integer rectifiable varifolds in open sets. result The support of stationary varifolds is C∞ rectifiable. This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space satisfying integrability conditions on their first variation. Firstly, the study of pointwise power decay rates almost everywhere of the quadratic tilt-excess is completed by establishing the precise decay rate for two-di…
Study shows distance to boundary is always attained on varifolds with bounded curvature.
problem Understanding varifolds with bounded mean curvature in Riemannian manifolds.
method Proves a barrier principle at infinity using sharp maximum principles.
result Distance to boundary is always attained on varifolds with bounded curvature.
Generic scarring occurs along stable minimal hypersurfaces in 3-7 dimensional manifolds.
problem Understanding scarring behavior of minimal hypersurfaces along stable ones.
method Analyzing a generic metric on a manifold to show scarring of minimal hypersurfaces.
result Closed, embedded minimal hypersurfaces scarring along stable ones, with diverging area and Morse index.
The paper proves the regularity and compactness of stable CMC integral varifolds in codimension 1.
problem Stable CMC integral varifolds of codimension 1.
method Structural conditions and variational hypotheses.
result The support of the varifold is an immersed constant-mean-curvature (CMC) hypersurface under given conditions.
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 C1,α regularity for varifolds close to specific cones. Constructs minimal immersions with singularities.
problem Minimal immersions with singularities in metric spaces.
method Constructs minimal immersions with catenoidal necks or floating disks converging to a singular point.
result Constructs minimal immersions with singularities.
Study phase transitions with prescribed mean curvature in Riemannian manifolds.
problem Understanding phase transitions with prescribed mean curvature in geometric settings.
method Analyzing solutions to inhomogeneous semilinear elliptic PDEs, establishing bounds and asymptotics.
result Established upper and lower bounds for eigenvalues of phase transition problems.