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Alexis Michelat

Publications and source records attributed to Alexis Michelat.

16 recordsLinked to original sources
arXiv2025

Morse Index Stability of Branched Willmore Immersions

We show that the sum of the Morse index and the nullity of Willmore immersions of bounded energy is lower semi-continuous without assuming that the limiting immersion and the bubbles are free of branch points. Our proof is based on a refined analysis of the properties of two families of fourth-order differential operators with regular singularities that depend on a parameter equal to the order of the branch points. The most technical results that justify the length of the article are Gagliardo-Nirenberg-Rellich inequalities in degenerating annuli that are necessary to show that the eigenvalues of the index operator with respect to a suitable weight are bounded from below.

math.DG↗
arXiv2023

Morse Index Stability of Biharmonic Maps in Critical Dimension

Furthering the development of Da Lio-Gianocca-Rivi\`ere's Morse stability theory (arXiv:2212.03124) that was first applied to harmonic maps between manifolds and later extended to the case of Willmore immersions (arXiv:2306.04608-04609), we generalise the method to the case of (intrinsic or extrinsic) biharmonic maps. In the course of the proof, we develop a novel method to prove strong energy quantization (in the space of squared-integrable functions that corresponds to the pre-dual of the Marcinkiewicz space of weakly squared-integrable functions) in a wide class of problems in geometric analysis, which allows us to recover some previous results in a unified fashion.

math.AP↗
arXiv2023

Morse Index Stability of Willmore Immersions I

In a recent work, F. Da Lio, M. Gianocca, and T. Rivi\`ere developped a new method to show upper semi-continuity results in geometric analysis, which they applied to conformally invariant Lagrangians in dimension $2$ (that include harmonic maps). In this article, we apply this method to show that the sum of the Morse index and the nullity of Willmore immersions is upper semi-continuous, provided that the limiting immersions and the bubbles are free of branch points. Our result covers the case of degenerating Riemann surfaces for which the ratio of the second residue (considered by P. Laurain and T. Rivi\`ere in their work on the energy quantization of Willmore surfaces) and the length of the minimal shrinking geodesic of the underlying sequence of Riemann surfaces is smaller than a universal constant.

math.DG↗
arXiv2023

Weighted Eigenvalue Problems for Fourth-Order Operators in Degenerating Annuli

We obtain a nigh optimal estimate for the first eigenvalue of two natural weighted problems associated to the bilaplacian (and of a continuous family of fourth-order elliptic operators in dimension $2$) in degenerating annuli (that are central objects in bubble tree analysis) in all dimension. The estimate depends only on the conformal class of the annulus. We also show that in dimension $2$ and dimension $4$, the first eigenfunction (of the first problem) is never radial provided that the conformal class of the annulus is large enough. The other result is a weighted Poincar\'e-type inequality in annuli for those fourth-order operators. Applications to Morse theory are given.

math.AP↗
arXiv2023

The Loewner Energy via the Renormalised Energy of Moving Frames

We obtain a new formula for the Loewner energy of Jordan curves on the sphere, which is a Kähler potential for the essentially unique Kähler metric on the Weil-Petersson universal Teichmüller space, as the renormalised energy of moving frames on the two domains of the sphere delimited by the given curve.

math.DG↗
arXiv2021

Quantization of the Willmore Energy in Riemannian Manifolds

We show that the quantization of energy for Willmore spheres into closed Riemannian manifolds holds provided that the Willmore energy and the area are uniformly bounded. The analogous energy quantization result holds for Willmore surfaces of arbitrary genus, under the additional assumptions that the immersion maps weakly converge to a limiting (possibly branched, weak immersion) map from the same surface, and that the conformal structures stay in a compact domain of the moduli space.

math.AP↗
arXiv2019

On the Morse Index of Branched Willmore Spheres in $3$-Space

We develop a general method to compute the Morse index of branched Willmore spheres and show that the Morse index is equal to the index of certain matrix whose dimension is equal to the number of ends of the dual minimal surface. As a corollary, we find that for all immersed Willmore spheres $\vecΦ:S^2\rightarrow \mathbb{R}^3$ such that $W(\vecΦ)=4πn$, we have $\mathrm{Ind}_{W}(\vecΦ)\leq n-1$

math.DG↗
arXiv2019

On the Moduli Space of Null Curves in Klein's Quadric

We study the moduli space of null curves in Klein's quartic in the four-dimensional (complex) projective plane using methods developed by Robert Bryant. As a consequence, we show that minimal surfaces with $9$ embedded planar ends do not exist and formulate some conjectures about the previous moduli space.

math.DG↗
arXiv2019

The Classification of Branched Willmore Spheres in the $3$-Sphere and the $4$-Sphere

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 branched Willmore spheres in $S^4$, generalising a theorem of Sebástian Montiel. As a result of our asymptotic analysis at branch points, we obtain an improved $C^{1,1}$ regularity of the unit normal of variational branched Willmore surfaces in arbitrary codimension. We also prove that the width of Willmore sphere min-max procedures in dimension $3$ and $4$, such as the sphere eversion, is an integer multiple of $4π$.

math.DG↗
arXiv2019

Higher Regularity of Weak Limits of Willmore Immersions I

We obtain in arbitrary codimension a removability result on the order of singularity of weak limits and bubbles of Willmore immersions measured by the second residue. This permits to reduce significantly the number of possible bubbling scenarii. 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 may occur as conformal images of bubbles of Willmore immersions.

math.AP↗
arXiv2019

Higher Regularity of Weak Limits of Willmore Immersions II

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 may occur as conformal images of bubbles in Willmore min-max problems.

math.AP↗
arXiv2018

Morse Index Estimates of Min-Max Willmore Surfaces

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 the others must be stable.

math.DG↗
arXiv2018

On the Morse Index of Critical Points in the Viscosity Method

We show that in viscous approximations of functionals defined on Finsler manifolds, it is possible to construct suitable sequences of critical points of these approximations satisfying the expected Morse index bounds as in Lazer-Solimini's theory, together with the entropy condition of Michael Struwe.

math.AP↗
arXiv2016

Morse Index of Willmore spheres in $S^3$

We obtain an upper bound for the Morse index of Willmore spheres $Σ\subset S^3$ coming from an immersion of $S^2$. The quantization of Willmore energy shows that there exists an integer $m$ such that $\mathscr{W}(Σ)=4πm$. Then we show that $\mathrm{Ind}_{\mathscr{W}}(Σ)\leq m$. The proof relies on an explicit computation relating the second derivative of $\mathscr{W}$ for $Σ$ with the Jacobi operator of the minimal surface in $\mathbb{R}^3$ it is the image of by stereographic projection thanks of the fundamental classification of Robert Bryant.

math.DG↗
arXiv2015

A Viscosity Method for the Min-Max Construction of Closed Geodesics

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 of our sequence of critical points converging towards a closed non-trivial geodesic.

math.AP↗