arXiv research

Anisotropic min-max theory constructs stable minimal surfaces in 3-manifolds.

Variational method for eigenvalues on manifolds.

Equity-Transformer solves NP-hard min-max routing problems efficiently.

We present a viscosity approach to the min-max construction of closed geodesics on compact Riemannian manifolds of arbitrary dimension. We also construct counter-examples in dimension $1$ and $2$ to the $\varepsilon$-regularity in the convergence procedure. Furthermore, we prove the lower semi-continuity of the index o…

The study bounds the topology of free boundary minimal surfaces in 3D manifolds.

Proves existence of special 2-spheres in curved 3-spaces.

In this paper we prove genus bounds for closed embedded minimal surfaces in a closed 3-dimensional manifold constructed via min-max arguments. A stronger estimate was announced by Pitts and Rubistein but to our knowledge its proof has never been published. Our proof follows ideas of Simon and uses an extension of a fam…

In this paper, we consider first-order convergence theory and algorithms for solving a class of non-convex non-concave min-max saddle-point problems, whose objective function is weakly convex in the variables of minimization and weakly concave in the variables of maximization. It has many important applications in mach…

Entropy is a natural geometric quantity measuring the complexity of a surface embedded in $\mathbb{R}^3$. For dynamical reasons relating to mean curvature flow, Colding-Ilmanen-Minicozzi-White conjectured that the entropy of any closed surface is at least that of the self-shrinking two-sphere. We prove this conjecture …

We use the theory of rectifiable metric spaces to define a Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative, or equivalently of the approximate local dilatation, of the Lipschitz functions. …

The paper generates future brain imaging sequences for Alzheimer's disease detection.

We prove the energy identity for min-max sequences of the Sacks-Uhlenbeck and the biharmonic approximation of harmonic maps from surfaces into general target manifolds. The proof relies on Hopf-differential type estimates for the two approximations and on estimates for the concentration radius of bubbles.

The paper proves the existence of boundary minimal hypersurfaces in compact manifolds with boundary.

Existence proof of noncompact self-shrinkers with arbitrary genus.

New insights into $p$-widths of surfaces, proving optimality and calculating constants.

The paper constructs new non-trivial harmonic maps into higher-dimensional target manifolds.

Study shows strong min-max principle for phase transitions.

We propose a principled method for kernel learning, which relies on a Fourier-analytic characterization of translation-invariant or rotation-invariant kernels. Our method produces a sequence of feature maps, iteratively refining the SVM margin. We provide rigorous guarantees for optimality and generalization, interpret…

Upper bound for Morse index of min-max varifolds.

Localized min-max method proves minimal hypersurface existence.

The paper solves min-max widths on a 3-sphere and strengthens multiplicity theorems.

Study introduces statistical mechanics for min-max problems.

Adaptive momentum method solves non-convex min-max problems.

Paper proves finiteness and Morse index estimates for equivariant min-max hypersurfaces.

New proof of Smale conjecture for RP^3 and lens spaces using min-max theory.

Paper improves Morse index bound for hypersurfaces.

New methods solve min-max problems on manifolds using Riemannian Hamiltonians.

Bound on equivariant index for min-max surfaces.

We prove that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves, are two-sided and have multiplicity one. This confirms a conjecture by Marques-Neves. We prove that in a bumpy metric each…

Survey of advances in non-convex min-max optimization for applications.

We reformulate LIPs as min-max problems for easier solution.

The min-max problem, also known as the saddle point problem, is a class of optimization problems which minimizes and maximizes two subsets of variables simultaneously. This class of problems can be used to formulate a wide range of signal processing and communication (SPCOM) problems. Despite its popularity, most exist…

The paper bounds the min-max width of embedded circles on spheres and manifolds.

Study confirms a 2-sphere metric with three geodesics of minimal length.

Bayesian optimization methods improved for min max optimization problems.

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts-Schoen-Simon \cite{AF62, AF65, P81, SS81} in a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature for all dimensions. The min-max hypersurface has a singular set of Hausdorff codimension $7$. We characterize the …

The study proves a generic multiplicity one theorem for $G$-invariant minimal hypersurfaces.

Constructs cmc doublings of minimal surfaces via min-max theory.

This research proves that two min-max theories for hypersurfaces are equivalent.

This study reveals a Min-Max property in LeNet's convolutional layers, enhancing adversarial robustness.

New Gaussian min-max theorem extends classical results to non-i.i.d. Gaussian matrices.

New algorithm solves non-convex, non-differentiable min-max games.

Generic density of equivariant min-max hypersurfaces in Riemannian manifolds.

We characterize the Zoll Riemannian metrics on a given simply connected spin closed manifold as those Riemannian metrics for which two suitable min-max values in a finite dimensional loop space coincide. We also show that on odd dimensional Riemannian spheres, when certain pairs of min-max values in the loop space coin…

New geometric invariant from min-max width of spheres on Riemannian 2-spheres.

Given a Riemannian manifold and a closed submanifold, we find a geodesic segment with free boundary on the given submanifold. This is a corollary of the min-max theory which we develop in this article for the free boundary variational problem. In particular, we develop a modified Birkhoff curve shortening process to ac…

In this paper, we develop a min-max theory for the construction of constant mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary closed manifold. As a corollary, we prove the existence of a nontrivial, smooth, closed, almost embedded, CMC hypersurface of any given mean curvature $c$. Moreover…

The study finds anisotropic minimal surfaces in 3-manifolds with smooth boundaries.