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Tristan Rivière

Publications and source records attributed to Tristan Rivière.

At least 37 records · Page 2Linked to original sources
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 Poincaré-type inequality in annuli for those fourth-order operators. Applications to Morse theory are given.

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arXiv2023

Minmax Hierarchies, Minimal Fibrations and a PDE based Proof of the Willmore Conjecture

We introduce a general scheme that permits to generate successive min-max problems for producing critical points of higher and higher indices to Palais-Smale Functionals in normal Banach manifolds equipped with complete Finsler structures. We call the resulting tree of minmax problems a minmax hierarchy. We give several examples and in particular we explain how to implement this scheme in the framework of the viscosity method introduced by the author some years ago in order to give a new proof of the Willmore conjecture after the famous result by Marques and Neves.

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arXiv2023

Morse Index Stability for Critical Points to Conformally invariant Lagrangians

We prove the upper-semi-continuity of the Morse index plus nullity of critical points to general conformally invariant Lagrangians in dimension 2 under weak convergence. Precisely we establish that the sum of the Morse indices and the nullity of an arbitrary sequence of weakly converging critical points to a general conformally invariant Lagrangians of maps from an arbitrary closed surface into an arbitrary closed smooth manifold passes to the limit in the following sense : it is asymptotically bounded from above by the sum of the Morse indices plus the nullity of the weak limit and the bubbles, while it was well known that the sum of the Morse index of the weak limit with the Morse indices of the bubbles is asymptotically bounded from above by the Morse indices of the weakly converging sequence. The main result is then extended to the case of sequences of maps from sequences of domains degenerating to a punctured Riemann surface assuming that the lengths of the images by the maps of the collars associated to this degeneration stay below some critical length.

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arXiv2022

The Unique Tangent Cone Property for Weakly Holomorphic Maps into Projective Algebraic Varieties

In the present paper, we establish the uniqueness of tangent maps for general weakly holomorphic and locally approximable maps from an arbitrary almost complex manifold into projective algebraic varieties. As a byproduct of the approach and the techniques developed we also obtain the unique tangent cone property for a special class of non-rectifiable positive pseudo-holomorphic cycles. This approach gives also a new proof of the main result by C. Bellettini on the uniqueness of tangent cones for positive integral $(p,p)$-cycles in arbitrary almost complex manifolds.

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arXiv2022

The Parametric Approach to the Willmore Flow

We introduce a parametric framework for the study of Willmore gradient flows which enables to consider a general class of weak, energy-level solutions and opens the possibility to study energy quantization and finite-time singularities. We restrict in this first work to a small-energy regime and prove that, for small-energy weak immersions, the Cauchy problem in this class admits a unique solution.

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arXiv2021

Integrability by compensation for Dirac Equation

We consider the Dirac Operator acting on the Clifford Algebra ${C\ell}_{m}$. We show that under critical assumptions on the potential and the spinor field the equation is subject to an integrability by compensation phenomenon and has a sub-critical behaviour below some positive energy threshold (i.e. $ε-$regularity theorem). This extends in 4 space dimensions as well as in 3 dimensions a similar result obtained previously by the two first authors in 2 D in \cite{DLR1}.

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arXiv2021

Almost Monotonicity Formula for H-minimal Legendrian Surfaces in the Heisenberg Group

We prove an almost monotonicity formula for H-minimal Legendrian Surfaces (also called {\it contact stationary legendrian immersions}) in the Heisenberg Group ${\mathbb H}^2$ . From this formula we deduce a Bernstein-Liouville type theorem for H-minimal Legendrian Surfaces. This last result happens to be in particular a central ingredient in the analysis part of the proof of the Willmore conjecture given by the author in arXiv:2007.05467

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arXiv2020

Bergman-Bourgain-Brezis-type Inequality

In this note, we prove a fractional version in $1$-D of the Bourgain-Brezis inequality \cite{bourgain1}. We show that such an inequality is equivalent to the fact that a holomorphic function $f\colon\D\to\C$ belongs to the Bergman space ${\mathcal{A}}^2(\D)$, namely $f\in L^2(\D)$, if and only if $$\|f\|_{ L^1+ {H}^{-1/2}(S^1)}:=\limsup_{r\to 1^-}\|f(re^{iθ})\|_{ L^1+ {H}^{-1/2}(S^1)}<+\infty.$$ Possible generalisations to the higher-dimensional torus are explored.

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arXiv2020

Critical Chirality in Elliptic Systems

We establish the regularity in 2 dimensions of $L^2$ solutions to critical elliptic systems in divergence form involving involution operators of finite $W^{1,2}$-energy.

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arXiv2019

Weak Holomorphic Structures over Kähler Surfaces

In this work we prove that any unitary Sobolev $W^{1,2}$ connection of an Hermitian bundle over a 2-dimensional Kähler manifold whose curvature is $(1,1)$ defines a smooth holomorphic structure. We prove moreover that such a connection can be strongly approximated in any $W^{1,p}$ ($p<2$) norm by smooth connections satisfying the same integrability condition.

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arXiv2019

3-Commutators Revisited

We present a class of Pseudo-differential elliptic systems with anti-self-dual potentials on ${\mathbb R}$ satisfying compensation phenomena similar to the ones for elliptic systems with anti-symmetric potentials. These compensation phenomena are based on new "multi-commutator" structures generalizing the 3-commtators introduced by the authors in a previous work some years ago.

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arXiv2019

Existence of infinitely many minimal hypersurfaces in low dimensions, after F.C. Marques, A.A. Neves et A. Song (Bourbaki Seminar)

A classical result by Marston Morse asserts that on some ellipsoids of ${\mathbb R}^3$ there exists exactly 3 closed and simple geodesics. The goal of this presentation is to prove that this rigidity result does not extend to higher dimensions and, more precisely, on any smooth closed riemannian manifod of arbitrary dimension between 3 and 7 there exists infinitely many closed embedded minimal surfaces. We are going to present the origins of this theorem as well as it's proof given recently by Antoine Song.

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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 $4π$.

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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.

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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.

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arXiv2018

The Space of Weak Connections in High Dimensions

The space of Sobolev connections, as it has been introduced for studying the variation of Yang-Mills Lagrangian in the critical dimension $4$, happens not to be weakly sequentially complete in dimension larger than $4$. This is a major obstruction for studying the variations of this important Lagrangian in high dimensions. The present paper generalizes the previous result of the authors from 'The resolution of the Yang-Mills Plateau problem in super-critical dimensions', Adv. Math. 316 (arxiv:1306.2010), valid in $5$ dimensions to arbitrary dimension and introduces a space of so called 'weak connections' for which we prove the weak sequential closure under Yang-Mills energy control. We also establish a strong approximation property of any weak connection by smooth connections away from codimension $5$ polyhedral sets. This last property is used in a subsequent work in preparation for establishing the partial regularity property for general stationary Yang-Mills weak connections.

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