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André Neves

Publications and source records attributed to André Neves.

30 records · Page 2Linked to original sources
arXiv2016

The catenoid estimate and its geometric applications

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 manifolds can be isotoped to be minimal and iii) the "doublings" of the Clifford torus by Kapouleas-Yang can be constructed variationally by an equivariant min-max procedure. In higher dimensions we also prove that the width of manifolds with positive Ricci curvature is achieved by an index 1 orientable minimal hypersurface.

math.DG↗
arXiv2015

Min-max theory and the energy of links

Freedman, He, and Wang, conjectured in 1994 that the Mobius energy should be minimized, among the class of all nontrivial links in Euclidean space, by the stereographic projection of the standard Hopf link. We prove this conjecture using the min-max theory of minimal surfaces.

math.GT↗
arXiv2014

New applications of Min-max Theory

I will talk about my recent work with Fernando Marques where we used Almgren-Pitts Min-max Theory to settle some open questions in Geometry: The Willmore conjecture, the Freedman-He-Wang conjecture for links (jointly with Ian Agol), and the existence of infinitely many minimal hypersurfaces in manifolds of positive Ricci curvature. Some open questions are suggested in the last section.

math.DG↗
arXiv2014

The Willmore conjecture

The Willmore conjecture, proposed in 1965, concerns the quest to find the best torus of all. This problem has inspired a lot of mathematics over the years, helping bringing together ideas from subjects like conformal geometry, partial differential equations, algebraic geometry and geometric measure theory. In this article we survey the history of the conjecture and our recent solution through the min-max approach. We finish with a discussion of some of the many open questions that remain in the field.

math.DG↗
arXiv2013

The Penrose inequality for asymptotically locally hyperbolic spaces with nonpositive mass

In the asymptotically locally hyperbolic setting it is possible to have metrics with scalar curvature at least -6 and negative mass when the genus of the conformal boundary at infinity is positive. Using inverse mean curvature flow, we prove a Penrose inequality for these negative mass metrics. The motivation comes from a previous result of P. Chruściel and W. Simon, which states that the Penrose inequality we prove implies a static uniqueness theorem for negative mass Kottler metrics.

math.DG↗
arXiv2013

Min-Max theory and the Willmore conjecture

In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in Euclidean three-space is at least 2π^2. We prove this conjecture using the min-max theory of minimal surfaces.

math.DG↗
arXiv2012

Uniqueness of Lagrangian Self-Expanders

We show that zero-Maslov class Lagrangian self-expanders in C^n which are asymptotic to a pair of planes intersecting transversely are locally unique if n>2 and unique if n=2.

math.DG↗
arXiv2012

Finite Time Singularities for Lagrangian Mean Curvature Flow

Given any embedded Lagrangian on a four dimensional compact Calabi-Yau, we find another Lagrangian in the same Hamiltonian isotopy class which develops a finite time singularity under mean curvature flow. This contradicts a weaker version of the Thomas-Yau conjecture regarding long time existence and convergence of Lagrangian mean curvature flow.

math.DG↗
arXiv2007

Translating solutions to Lagrangian mean curvature flow

We prove some non-existence theorems for translating solutions to Lagrangian mean curvature flow. More precisely, we show that translating solutions with an $L^2$ bound on the mean curvature are planes and that almost-calibrated translating solutions which are static are also planes. Recent work of D. Joyce, Y.-I. Lee, and M.-P. Tsui, shows that these conditions are optimal.

math.DG↗
arXiv2006

3-Manifolds with Yamabe invariant greater than that of $\RP^3$

We complete the classification (started by Bray and the second author) of all closed 3-manifolds with Yamabe invariant greater than that of $\RP^3$, by showing that such manifolds are either $S^3$ or finite connected sums $# m(S^2 \times S^1) # n(S^2 \tilde{\times} S^1)$ for $m + n \geq 1$, where $S^2 \tilde{\times} S^1$ is the nonorientable $S^2$-bundle over $S^1$. A key ingredient is Aubin's Lemma, which says that if the Yamabe constant is positive, then it is strictly less than the Yamabe constant of any of its non-trivial finite conformal coverings. This lemma, combined with inverse mean curvature flow and with analysis of the Green's functions for the conformal Laplacians on specific finite and normal infinite Riemannian coverings, will allow us to construct a family of nice test functions on the finite coverings and thus prove the desired result.

math.DG↗