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.
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.
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.
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.
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…
In this work the Isoperimetric Inequality for integral varifolds is used to obtain sharp estimates for the size of the set where the density quotient is small and to generalise Calderón's and Zygmund's theory of first order differentiability for functions in Lebesgue spaces from Lebesgue measure to 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…
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…
We survey - by means of 20 examples - the concept of varifold, as generalised submanifold, with emphasis on regularity of integral varifolds with mean curvature, while keeping prerequisites to a minimum. Integral varifolds are the natural language for studying the variational theory of the area integrand if one conside…
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.
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.
Proves singular set of certain integral hypercurrents has measure zero.
problem Characterizing singular sets of specific integral hypercurrents.
method Proof based on varifold stationarity.
result Singular set has measure zero.
Proves regularity for stable varifolds near cones, expanding previous work.
problem Understanding stable varifolds near singularities.
method Proves C1,α regularity for varifolds in SQ. result Furthers understanding of local structure near singularities.
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. 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.
In this work a local inequality is provided which bounds the distance of an integral varifold from a multivalued plane (height) by its tilt and mean curvature. The bounds obtained for the exponents of the Lebesgue spaces involved are shown to be sharp.
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.
We give a necessary and sufficient geometric structural condition for a stable codimension 1 integral varifold on a smooth Riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable singularities; when this condition is satisfied, the singular set is empty if the…
The paper proves regularity for varifolds with bounded anisotropic mean curvature.
problem Regularity of varifolds with bounded anisotropic mean curvature.
method Local anisotropic regularity theorem and touching balls approach.
result Varifolds can be covered by countably many C2-regular submanifolds. 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…
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.
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.
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 give two structural conditions on a codimension 1 integral n-varifold with first variation locally summable to an exponent p>n that imply the following: whenever each orientable portion of the C1-embedded part of the varifold (which is non-empty by the Allard regularity theory) is stationarity and the $C^…
We prove existence and partial regularity of integral rectifiable m-dimensional varifolds minimizing functionals of the type ∫∣H∣p and ∫∣A∣p in a given Riemannian n-dimensional manifold (N,g), 2≤m<n and p>m, under suitable assumptions on N (in the end of the paper we give many examples of …
Study bounds singular set of minimal hypersurfaces with index control.
problem Estimating singular set size of minimal hypersurfaces.
method Finite index and null singular set conditions on integral varifolds.
result Local measure bounds on singular set and upper Minkowski content.
The paper proves rigidity estimates for varifolds almost minimizing the Willmore energy.
problem Optimal rigidity estimates for varifolds almost minimizing the Willmore energy.
method Analyzes integral 2-varifolds with generalized mean curvature in Rn. result Varifolds are close to the standard embedding of the round sphere in a quantitative way.
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…
Compact capillary hypersurfaces with specific curvatures are proven.
problem Proving compactness of capillary hypersurfaces with prescribed mean curvature.
method Extending curvature varifolds to oriented integral varifolds and applying curvature varifolds with capillary boundary.
result Compactness result for capillary hypersurfaces with mean curvature prescribed by ambient functions.
Based on a novel type of Sobolev-Poincaré inequality (for generalised weakly differentiable functions on varifolds), we establish a finite upper bound of the geodesic diameter of generalised compact connected surfaces-with-boundary of arbitrary dimension in Euclidean space in terms of the mean curvatures of the surface…
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 C1,γ hypersurface near the boundary. New, shorter proofs for varifolds and flows with improved decay of flatness.
problem Proving regularity theorems for varifolds and flows with bounded first variation and forcing.
method Decay of flatness via weighted monotonicity formulas and viscosity approach.
result Improved proofs with decay of flatness and characterization of blow-ups.
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.
Constructs area-minimizing submanifolds with fractal singularities.
problem Area-minimizing submanifolds with fractal singular sets.
method Integral currents, mod v currents, stable stationary varifolds.
result Sharp dimensionwise solution to Almgren's conjecture.
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 L2 curvature in the integer rectifiable class. In this paper we deal with singular varieties of bounded mean curvature in the viscosity sense. They contain all varifolds of bounded generalized mean curvature. In the first part we investigate the second-order properties of these varieties, obtaining results that are new also in the varifold's setting. In particular …
In a previous paper we developed a regularity and compactness theory in Euclidean ambient spaces for codimension 1 weakly stable CMC integral varifolds satisfying two (necessary) structural conditions. Here we generalize this theory to the setting where the mean curvature (of the regular part of the varifold) is prescr…
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 B1/2(0), V is represented by the graph of a Lipschitz 2-valued function over P0 with small Lipschitz constant; all tangent cones at singular points are unique and comprised of stationary unions of 4 half-planes. Proves branch set dimension for stationary varifolds with ε-regularity.
problem Analyzing the structure of stationary varifolds with ε-regularity.
method Utilizes planar frequency function and geometric analysis.
result Hausdorff dimension of branch set is at most n-2 for certain varifolds.
The Clifford torus minimizes Willmore energy closely for small perturbations.
problem Finding the closest shape to the Clifford torus under small perturbations of Willmore energy.
method Analyzing integral 2-varifolds with specific properties and showing quantitative closeness to the Clifford torus.
result The support of the varifold is quantitatively close to the Clifford torus after a conformal transformation.
Upper bound found for minimal area in Einstein 4-manifolds.
problem Finding the smallest area of a 2D varifold in closed Einstein 4-manifolds.
method Using homological filling functions and quantitative Sobolev and ε-regularity constants for Einstein metrics.
result An upper bound A(M,g)≤FEin(v,D) for the area of 2D varifolds in Einstein 4-manifolds. Stable nets on convex hypersurfaces maintain their shape under small perturbations.
problem Maintaining the shape of nets on convex surfaces under slight changes.
method Constructing stable geodesic nets on convex hypersurfaces.
result Stable geodesic nets on convex hypersurfaces do not change shape under small perturbations.
Study min-max theory for hypersurfaces with boundary constraints.
problem Finding minimal hypersurfaces with boundary constraints.
method Schoen-Simon-type regularity result for integral varifolds, proving existence of closed hypersurfaces with specific properties.
result Existence of a closed C1,1 hypersurface with codimension ≥7 singular set in the interior. 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.
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 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.
Study shows persistence of singularities in minimal hypersurfaces is rare.
problem Persistence of singularities in minimal hypersurfaces.
method Analysis of Fredholm index of the Jacobi operator and Morse indices of conical links.
result Only non-strongly isolated singularities may persist in generic metrics.