arXiv research

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

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.

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

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

Proves conjecture about sphere widths under rotational symmetry.

Upper bound for Morse index of min-max varifolds.

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

Study finds geodesic networks for surfaces with convex boundary.

How large can be the width of Riemannian three-spheres of the same volume in the same conformal class? If a maximum value is attained, how does a maximising metric look like? What happens as the conformal class changes? In this paper, we investigate these and other related questions, focusing on the context of Simon-Sm…

We characterize the first min-max width of real projective spaces of any dimension. The width is the minimum area over the Clifford hypersurfaces. We also compute the Morse index of the Clifford hypersurfaces in the complex and quaternionic projective spaces.

We compute the $k$-width of a round $2$-sphere for $k=1,\ldots,8$ and we use this result to show that unstable embedded closed geodesics can arise with multiplicity as a min-max critical varifold.

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

We show that the sum of the Morse indices of the Willmore spheres realising the width of Willmore type sweep-outs is bounded by the number of the parameters of the min-max. As an application, we deduce that among the true Willmore spheres realising the min-max sphere eversion, at most one of them one has index 1, while…

In this paper, we show that a closed manifold $M^{n+1} (n \geq 7)$ endowed with a $C^\infty$-generic (Baire sense) metric contains infinitely many singular minimal hypersurfaces with optimal regularity. Moreover, for $2 \leq n \leq 6$, our argument also implies the denseness of the minimal hypersurfaces realizing min-m…

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

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

The paper develops a theory for free boundary minimal surfaces with genus at least one.

Given any admissible $k$-dimensional family of immersions of a given closed oriented surface into an arbitrary closed Riemannian manifold, we prove that the corresponding min-max width for the area is achieved by a smooth (possibly branched) immersed minimal surface with multiplicity one and Morse index bounded by $k$.

We prove a sharp area estimate for catenoids that allows us to rule out the phenomenon of multiplicity in min-max theory in several settings. We apply it to prove that i) the width of a three-manifold with positive Ricci curvature is realized by an orientable minimal surface ii) minimal genus Heegaard surfaces in such …

The paper finds a special hypersurface in a manifold with positive Ricci curvature.

Study min-max theory for hypersurfaces with boundary constraints.

We perform a replacement procedure in order to produce a free boundary minimal surface whose area achieves the min-max value over all disk sweepouts of a manifold whose boundary lie in a submanifold. Our result is based on a proof of the convexity of the energy for free boundary harmonic maps and a generalization of Co…

We obtain in arbitrary codimension a removability result on the order of singularity of Willmore surfaces realising the width of Willmore min-max problems on spheres. As a consequence, out of the twelve families of non-planar minimal surfaces in $\mathbb{R}^3$ of total curvature greater than $-12π$, only three of them …

In this paper we consider min-max minimal surfaces in three-manifolds and prove some rigidity results. For instance, we prove that any metric on a 3-sphere which has scalar curvature greater than or equal to 6 and is not round must have an embedded minimal sphere of area strictly smaller than $4π$ and index at most one…

The paper explains how simple methods can converge to optimal solutions in complex neural games.

Minimum width for ReLU networks on compact domain is exactly max{d_x, d_y, 2}

The paper proves the existence of $G$-invariant minimal hypersurfaces on certain Riemannian manifolds.

Let $M^{n+1}$ be an orientable compact Riemannian manifold with positive Ricci curvature. We prove that the Almgren-Pitts width of $M^{n+1}$ is achieved by an orientable index $1$ minimal hypersurface with multiplicity $1$ and optimal regularity. This extends to dimensions $n+1\geq 8$ the results of Ketover-Marques-Nev…

Paper proves rigidity of minimal disks in 3-balls with non-negative Ricci curvature.

Rigidity theorem for critical points of Allen-Cahn equation on S³.

Adaptive kernels from neural networks improve model performance.

Our work proves convergence to low robust training loss for polynomial width ReLU networks.

We study global variational properties of the space of solutions to $-\varepsilon^2Δu + W'(u)=0$ on any closed Riemannian manifold $M$. Our techniques are inspired by recent advances in the variational theory of minimal hypersurfaces and extend a well-known analogy with the theory of phase transitions. First, we show t…

We establish a min-max estimate on the volume width of a closed Riemannian manifold with nonnegative Ricci curvature. More precisely, we show that every closed Riemannian manifold with nonnegative Ricci curvature admits a PL Morse function whose level set volume is bounded in terms of the volume of the manifold. As a c…

Study on ground states of semilinear elliptic equations with various potential wells.

We extend the classification of Robert Bryant of Willmore spheres in $S^3$ to variational branched Willmore spheres $S^3$ and show that they are inverse stereographic projections of complete minimal surfaces with finite total curvature in $\mathbb{R}^3$ and vanishing flux. We also obtain a classification of variational…

We prove that any manifold diffeomorphic to $S^3$ and endowed with a generic metric contains at least two embedded minimal two-spheres. The existence of at least one minimal two-sphere was obtained by Simon-Smith in 1983. Our approach combines ideas from min-max theory and mean curvature flow. We also establish the exi…

Study shows strong min-max principle for phase transitions.

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

Localized min-max method proves minimal hypersurface existence.

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.