arXiv research

The paper simplifies arguments for stationary varifolds results.

Green functions on stationary varifolds established with inequalities and convergence results.

The paper proves smoothness of stationary varifolds.

Upper bound for Morse index of min-max varifolds.

Establishes a boundary maximum principle for varifolds with fixed contact angle.

Proves singular set of certain integral hypercurrents has measure zero.

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 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 …

Characterizes critical points in convex double and triple bubbles.

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…

Proves regularity for stable varifolds near cones, expanding previous work.

Proves branch set dimension for stationary varifolds with ε-regularity.

Proves regularity for stable varifolds near specific cones.

The study examines stationary integral varifolds near multiplicity 2 planes, proving regularity under specific conditions.

New theory for area of Legendrian surfaces, proving smoothness and variational results.

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…

New moving plane method for varifolds promotes smoothness from boundary to interior.

Extends Campanato theory to multi-valued functions for geometric variational problems.

Develops a PDE approach to constructing nontrivial anisotropic surfaces.

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…

Study the singular limit of a boundary reaction equation, showing energy concentration and varifold support.

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 $L^q$ norm of the second fundamental form of the immersion times a "viscosity" parameter. …

Study shows non-uniqueness of Brakke flow near flat singular points.

Study local minimizers of Ginzburg-Landau functionals in high dimensions, showing energy measures converge to rectifiable measures.

Constructs area-minimizing submanifolds with fractal singularities.

Upper bound found for minimal area in Einstein 4-manifolds.

Study calculates first $p$-widths of unit disk.

We study the asymptotics as $p\uparrow 2$ of stationary $p$-harmonic maps $u_p\in W^{1,p}(M,S^1)$ from a compact manifold $M^n$ to $S^1$, satisfying the natural energy growth condition $\int_M|du_p|^p=O(\frac{1}{2-p}).$ Along a subsequence $p_j\to 2$, we show that the singular sets $Sing(u_{p_j})$ converge to the sup…

Study shows persistence of singularities in minimal hypersurfaces is rare.

Study bounds singular set of minimal hypersurfaces with index control.

Constructs minimal immersions with singularities.

For a sequence of immersed connected closed Hamiltonian stationary Lagrangian submaniolds in $\mathbb{C}^{n}$ with uniform bounds on their volumes and the total extrinsic curvatures, we prove that a subsequence converges either to a point or to a Hamiltonian stationary Lagrangian $n$-varifold locally uniformly in $C^{k…

The paper establishes structure theory for stable varifolds with applications to area minimising hypersurfaces.

If one considers an integral varifold $I^m\subseteq M$ 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 $\dim S^k\leq k$. In complete generality nothing else is known about …

We develop an equivariant min-max theory as proposed by Pitts-Rubinstein in 1988 and then show that it can produce many of the known minimal surfaces in $\mathbb{S}^3$ up to genus and symmetry group. We also produce several new infinite families of minimal surfaces in $\mathbb{S}^3$ proposed by Pitts-Rubinstein. These …

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…

Given a Hermitian line bundle $L\to M$ over a closed, oriented Riemannian manifold $M$, we study the asymptotic behavior, as $ε\to 0$, of couples $(u_ε,\nabla_ε)$ critical for the rescalings \begin{align*} &E_ε(u,\nabla)=\int_M\Big(|\nabla u|^2+ε^2|F_\nabla|^2+\frac{1}{4ε^2}(1-|u|^2)^2\Big) \end{align*} of the self-dua…

The paper defines and proves the existence of decompositions of integral varifolds.

New varifold example shows decomposition failure.

In this note we show that the recent dynamical stability result for small $C^1$-perturbations of strongly stable minimal submanifolds of C.-J. Tsai and M.-T. Wang directly extends to the enhanced Brakke flows of Ilmanen. We illustrate applications of this result, including a local uniqueness statement for strongly stab…

Graphs with bounded anisotropic mean curvature are regular almost everywhere.

We use min-max techniques to produce nontrivial solutions $u_ε:M\to \mathbb{R}^2$ of the Ginzburg-Landau equation $Δu_ε+\frac{1}{ε^2}(1-|u_ε|^2)u_ε=0$ on a given compact Riemannian manifold, whose energy grows like $|\logε|$ as $ε\to 0$. When the degree one cohomology $H^1_{dR}(M)=0$, we show that the energy of these s…

In a recent paper the author introduced a new method based on viscosity techniques for producing minimal surfaces by minmax arguments. The present work corresponds to the regularity part of the method. Precisely we establish that any weakly conformal $W^{1,2}$ map from a riemann surface $S$ into a closed oriented sub-m…

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…

Paper provides estimates for 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…

Generalizes Reilly inequality to varifolds and analyzes equality cases.

Counterexample and new proof for curvature varifolds.