arXiv research

Paper improves Morse index bound for hypersurfaces.

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

Proves existence of minimal surfaces with fixed boundary contact angle.

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

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

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

We prove optimal genus bounds for minimal surfaces arising from the min-max construction of Simon-Smith. This confirms a conjecture made by Pitts-Rubinstein in 1986.

In this paper we survey with complete proofs some well--known, but hard to find, results about constructing closed embedded minimal surfaces in a closed 3-dimensional manifold via min--max arguments. This includes results of J. Pitts, F. Smith, and L. Simon and F. Smith.

New examples of non-bumpy metrics on spheres and projective spaces with multiplicity.

Develops a PDE approach to constructing nontrivial anisotropic surfaces.

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…

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…

Proves existence of at least two minimal spheres in any 3D space.

3D spheres can't be swept by short curves, complicating geodesic length estimates.

The paper proves unique geodesics on hyperbolic surfaces and finds lower bounds.

Theory proves existence of hypersurfaces with prescribed curvature.

Theory for capillary surfaces in 3-manifolds with smooth boundary.

Constructs a unique surface in a ball with specific properties.

In this paper, we establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary established by Fraser. Our theory also includes as a special case the min-max theory …

Study on geodesics proving index and intersection bounds, with examples of multiplicity.

Proves multiplicity one for boundary minimal hypersurfaces in compact manifolds.

The paper proves existence of special surfaces in 3D manifolds with constant curvature.

New minimal hypersphere found in 4-sphere solving Bernstein problem.

Extends diameter bounds for submanifolds with boundary and minor curvature restrictions.

Study finds nontrivial $n$-harmonic maps from $S^n$ to closed manifolds.

Study bounds the Morse index of a special torus to 1.

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…

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

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. …

The paper constructs infinitely many surfaces with specific mean curvature.

We prove that, for a generic set of smooth prescription functions $h$ on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature $h$. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersur…

We prove that given a three manifold with an arbitrary metric $(M^3, g)$ of positive Ricci curvature, there exists a sweepout of $M$ by surfaces of genus $\leq 3$ and areas bounded by $C vol(M^3, g)^{2/3}$. We use this result to construct a sweepout of $M$ by 1-cycles of length at most $C vol(M^3, g)^{1/3}$. The sweepo…

The study finds a way to create minimal surfaces with specific shapes in 3-manifolds.

We use a min-max procedure on the Allen-Cahn energy functional to construct geodesics on closed, 2-dimensional Riemannian manifolds, as motivated by the work of Guaraco. Borrowing classical blowup and curvature estimates from geometric analysis, as well as novel Allen-Cahn curvature estimates due to Wang-Wei, we manage…

The paper generalizes free boundary min-max theory to equivariant settings.

In this paper, we prove that the Morse index of a multiplicity one, smooth, min-max minimal hypersurface is generically equal to the dimension of the homology class detected by the families used in the construction. This confirms part of the program (\cite{marques-icm}, \cite{marques-neves-cycles}, \cite{marques-neves-…

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…

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…

We employ min-max methods to construct uncountably many, geometrically distinct, properly embedded geodesic lines in any asymptotically conical surface of non-negative scalar curvature, a setting where minimization schemes are doomed to fail. Our construction provides control of the Morse index of the geodesic lines we…

Variational method for eigenvalues on manifolds.

Study shows strong min-max principle for phase transitions.

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

Upper bound for Morse index of min-max varifolds.

New minimal hypersurfaces in 4D sphere found.

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.

The paper tackles robust statistical methods using Wasserstein DRO formulations.