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. 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.
We prove that every stationary polyhedral varifold minimizes area in the following senses: (1) its area cannot be decreased by a one-to-one Lipschitz ambient deformation that coincides with the identity outside of a compact set, and (2) it is the varifold associated to a mass-minimizing flat chain with coefficients in …
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…
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.
For a given family of smooth closed curves γ1,...,γα⊂R3 we consider the problem of finding an elastic \emph{connected} compact surface M with boundary γ=γ1∪...∪γα. This is realized by minimizing the Willmore energy W on a suitable class of competitors. While the direct minimi…
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.
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 …
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. 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.
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. We address the minimization of the Canham-Helfrich functional in presence of multiple phases. The problem is inspired by the modelization of heterogeneous biological membranes, which may feature variable bending rigidities and spontaneous curvatures. With respect to previous contributions, no symmetry of the minimizers…
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.
We establish an optimal regularity result for parametrized two-dimensional stationary varifolds. Namely, we show that the parametrization map is a smooth minimal branched immersion and that the multiplicity function is constant. We provide some applications of this regularity result, especially in the calculus of varia…
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^…
In this paper, we will show that Hausdorff convergence and varifold convergence coincide on the class of almost minimal sets.
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.
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.
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. We adapt the method of Simon [JDG '93] to prove a C1,α-regularity theorem for minimal varifolds which resemble a cone C02 over an equiangular geodesic net. For varifold classes admitting a "no-hole" condition on the singular set, we additionally establish C1,α-regularity near the cone $\bf{C}_0^2 \ti…
We establish a boundary maximum principle for free boundary minimal submanifolds in a Riemannian manifold with boundary, in any dimension and codimension. Our result holds more generally in the context of varifolds.
This paper proves that classical minimal surfaces of arbitrary topological type with total boundary curvature at most 4πmust be smoothly embedded. Related results are proved for varifolds and for soap film surfaces.
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.
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.
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. 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 calculates first p-widths of unit disk.
problem Computing first p-widths of the unit disk. method Regularity result for integral 1-varifolds on compact 2-manifolds with convex boundary, applied to unit disk.
result Computed first p-widths for p=1,...,4. 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. …
For compact Riemannian manifolds with convex boundary, B.White proved the following alternative: Either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small singular set. There is the natural question if a similar result is true for submanif…
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.
The paper proves inequalities for submanifolds in Riemannian manifolds.
problem Proving geometric inequalities for submanifolds in Riemannian manifolds.
method Using Rauch's comparison theorem and first variation formula.
result General Li-Yau inequality applicable in bounded sectional curvature manifolds.
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.
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.
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.
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.
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.
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…
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.
The paper studies minimal surfaces in 3D spheres and balls, confirming conjectures and identifying new surfaces.
problem Understanding minimal surfaces in 3D spheres and balls with low area.
method Equivariant optimization of Laplace and Steklov eigenvalues to construct minimal surfaces of prescribed topology.
result Sharp area estimates and varifold limits for minimal surfaces in large topology regimes.
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.
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.
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.
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…
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.
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.