arXiv subjects

Tristan Rivière

Publications and source records attributed to Tristan Rivière.

At least 55 records · Page 3Linked to original sources

arXiv2018

A Resolution of the Poisson Problem for Elastic Plates

The Poisson problem consists in finding an immersed surface $Σ\subset\mathbb{R}^m$ minimising Germain's elastic energy (known as Willmore energy in geometry) with prescribed boundary, boundary Gauss map and area which constitutes a non-linear model for the equilibrium state of thin, clamped elastic plates originating from the work of S. Germain and S.D. Poisson or the early XIX century. We present a solution to this problem consisting in the minimisation of the total curvature energy $E(Σ)=\int_Σ|\operatorname{I\!I}_Σ|^2_{g_Σ}\,\mathrm{d}vol_Σ$ ($\operatorname{I\!I}_Σ$ is the second fundamental form of $Σ$), which is variationally equivalent to the elastic energy, in the case of boundary data of class $C^{1,1}$ and when the boundary curve is simple and closed. The minimum is realised by an immersed disk, possibly with a finite number of branch points in its interior, which is of class $C^{1,α}$ up to the boundary for some $0<α<1$, and whose Gauss map extends to a map of class $C^{0,α}$ up to the boundary.

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arXiv2018

Lower Semi-Continuity of the Index in the Visosity Method for Minimal Surfaces

The goal of the present work is twofold. First we prove the existence of an Hilbert Manifold structure on the space of immersed oriented closed surfaces with three derivatives in $L^2$ in an arbitrary sub-manifold $M^m$ of an euclidian space $R^Q$. Second, using this Hilbert manifold structure, we prove a lower semi continuity property of the index for sequences of conformal immersions, critical points to the viscous approximation of the area satisfying Struwe entropy estimate and bubble tree strongly converging in $W^{1,2}$ to a limiting minimal surface as the viscous parameter is going to zero.

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arXiv2018

The regularity of parametrized integer stationary varifolds in two dimensions

We establish an optimal regularity result for parametrized two-dimensional stationary varifolds. Namely, we show that the parametrization map is a smooth minimal branched immersion and that the multiplicity function is constant. We provide some applications of this regularity result, especially in the calculus of variations for the area functional.

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arXiv2017

Uniform Regularity Results for Critical and Subcritical Surface Energies

We establish regularity results for critical points to energies of immersed surfaces depending on the first and the second fundamental form exclusively. These results hold for a large class of intrinsic elliptic Lagrangians which are sub-critical or critical. They are derived using uniform $ε-$regularity estimates which do not degenerate as the Lagrangians approach the critical regime given by the Willmore integrand

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arXiv2017

A Viscosity Method in the Min-max Theory of Minimal Surfaces

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. This relaxation of the area functional satisfies the Palais-Smale condition for $q>2$. This permits to construct critical points of the relaxed Lagrangian using classical min-max arguments such as the mountain pass lemma. The goal of this work is to describe the passage to the limit when the "viscosity" parameter tends to zero. Under some natural entropy condition, we establish a varifold convergence of these critical points towards a parametrized integer stationary varifold realizing the min-max value. It is proved in a recent work in collaboration with A.Pigati that parametrized integer stationary varifold are given by smooth maps exclusively. As a consequence we conclude that every surface area minmax is realized by a smooth possibly branched minimal immersion.

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arXiv2017

Minmax Hierarchies and Minimal Surfaces in Manifolds

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 Banach manifolds equipped with Finsler structures. We call the resulting tree of minmax problems a minmax hierarchy. Using the viscosity approach to the minmax theory of minimal surfaces introduced by the author in a series of recent works, we explain how this scheme can be deformed for producing smooth minimal surfaces of strictly increasing area in arbitrary codimension. We implement this scheme to the case of the $3-$dimensional sphere. In particular we are giving a min-max characterization of the Clifford Torus and conjecture what are the next minimal surfaces to come in the $S^3$ hierarchy. Among other results we prove here the lower semi continuity of the Morse Index in the viscosity method below an area level.

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arXiv2017

The Regularity of Conformal Target Harmonic Maps

In a recent paper the author introduced a new method based on viscosity techniques for producing minimal surfaces by minmax arguments. The present work corresponds to the regularity part of the method. Precisely we establish that any weakly conformal $W^{1,2}$ map from a riemann surface $S$ into a closed oriented sub-manifold $N^n$ of an euclidian space ${\mathbb R}^m$ realizes a stationary varifold if and only if it is a smooth conformal harmonic map form $S$ into $N^n$.

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arXiv2016

Energy Quantization of Willmore surfaces at the boundary of the Moduli Space

We establish an energy quantization result for sequences of Willmore surfaces when the underlying sequence of Riemann surfaces is degenerating in the moduli space. we notably exhibit a new residue which quantifies the potential loss of energy in collar regions. Thanks to these residues, we also prove compactness of Willmore immersion with bounded conformal class and energy below $12π$.

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arXiv2016

A Pohozaev-type formula and Quantization of Horizontal Half-Harmonic Maps

In a recent paper the first and the third authors introduced the notion of horizontal α-harmonic map, with respect to a given C^1 planes distribution P_T on all R^m. The goal of this paper is to investigate compactness and quantization properties of sequences of horizontal 1/2- harmonic maps u_k in 1D. The quantization analysis is obtained through a precise asymptotic development of the energy of u_k in the neck regions and a subtle application of new Pohozaev-type formulae.

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arXiv2016

The resolution of the Yang-Mills Plateau problem in super-critical dimensions

We study the minimization problem for the Yang-Mills energy under fixed boundary connection in supercritical dimension $n\geq 5$. We define the natural function space A_{G} in which to formulate this problem in analogy to the space of integral currents used for the classical Plateau problem. The space A_{G} can be also interpreted as a space of weak connections on a "real measure theoretic version" of reflexive sheaves from complex geometry. We prove the weak closure result which ensures the existence of energy-minimizing weak connections in A_{G}. We then prove that any weak connection from A_{G} can be obtained as a L^2-limit of classical connections over bundles with defects. This approximation result is then extended to a Morrey analogue. We prove the optimal regularity result for Yang-Mills local minimizers. On the way to prove this result we establish a Coulomb gauge extraction theorem for weak curvatures with small Yang-Mills density. This generalizes to the general framework of weak $L^2$ curvatures previous works of Meyer-Rivière and Tao-Tian in which respectively a strong approximability property and an admissibility property were assumed in addition.

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arXiv2016

Horizontal $α$-Harmonic Maps

Given a $C^1$ planes distribution $P_T$ on all ${\mathbb R}^m$ we consider {\em horizontal $α$-harmonic maps}, $α\ge 1/2$, with respect to such a distribution. These are maps $u\in H^α({\mathbb R}^k,{\mathbb R}^m)$ satisfying $P_T\nabla u=\nabla u$ and $P_T(u)(-Δ)^αu=0$ in ${\mathcal D}'({\mathbb R}^k).$ If the distribution of planes is integrable then we recover the classical case of $α$-harmonic maps with values into a manifold. In this paper we shall focus our attention to the case $α=1/2$ in dimension $1$ and $α=2$ in dimension $2$ and we investigate the regularity of the {\em horizontal $α$-harmonic maps}. In both cases we show that such maps satisfy a Schrödinger type system with an antisymmetric potential, that permits us to apply the previous results obtained by the authors. Finally we study the regularity of {\em variational $α$-harmonic} maps which are critical points of $\|(-Δ)^{α/2} u\|^2_{L^2}$ under the constraint to be tangent (horizontal) to a given planes distribution. We produce a convexification of this variational problem which permits to write it's Euler Lagrange equations.

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arXiv2015

Willmore Minmax Surfaces and the Cost of the Sphere Eversion

We develop a general Minmax procedure in Euclidian spaces for constructing Willmore surfaces of non zero indices. We implement this procedure to the Willmore Minmax Sphere Eversion in the 3 dimensional euclidian space. We compute the cost of the Sphere eversion in terms of Willmore energies of Willmore Spheres in ${\R}^3$

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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.

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arXiv2015

The Variations of Yang-Mills Lagrangian

Yang-Mills theory is growing at the interface between high energy physics and mathematics. It is well known that Yang-Mills theory and Gauge theory in general had a profound impact on the development of modern differential and algebraic geometry. One could quote Donaldson invariants in four dimensional differential topology, Hitchin Kobayashi conjecture relating the existence of Hermitian-Einstein metric on holomorphic bundles over Kähler manifolds and Mumford stability in complex geometry or also Gromov Witten invariants in symplectic geometry...etc. While the influence of Gauge theory in geometry is quite notorious, one tends sometimes to forget that Yang-Mills theory has been also at the heart of fundamental progresses in the non-linear analysis of Partial Differential Equations in the last decades. The purpose of this mini-course is to present the variations of this important lagrangian. We shall raise analysis questions such as existence and regularity of Yang-Mills minimizers or such as the compactification of the moduli space of critical points to Yang-mills lagrangian in general. These notes correspond to a mini-course given by the author on the subject at the 9th summer school in Differential Geometry at the Korean Institute for Advanced Studies between june 23rd and june 27th 2014.

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arXiv2015

Exploring the Unknown: the Work of Louis Nirenberg on Partial Differential Equations

Partial differential equations are central objects in the mathematical modeling of natural and social sciences (sound propagation, heat diffusion, thermodynamics, electromagnetism, elasticity, fluid dynamics, quantum mechanics, population growth, finance...etc). They were for a long time restricted only to the study of natural phenomena or questions pertaining to geometry, before becoming over the course of time, and especially in the last century, a field in itself. The second half of the XXth century was the "golden age" of the exploration of partial differential equations from a theoretical perspective. The mathematical work of Louis Nirenberg since the early 1950s has to a large extent contributed to the growth of this fundamental area of human knowledge. The name Nirenberg is associated with many of the milestones in the study of PDEs. The award of the Abel Prize to Louis Nirenberg marks a special occasion for us to revisit the development of the field of PDEs and the work of one of the main actors of its exploration.

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arXiv2015

Blow-up analysis of a nonlocal Liouville-type equation

In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-Δ)^\frac12 u= κe^u-1~\mbox{in $S^1$,} \end{equation} where $(-Δ)^\frac{1}{2}$ stands for the fractional Laplacian and $κ$ is a bounded function. We interpret the above equation as the prescribed curvature equation to a curve in conformal parametrization. We also establish a relation between this equation and the analogous equation in $\mathbb{R}$ \begin{equation} (-Δ)^\frac{1}{2} u =Ke^u \quad \text{in }\mathbb{R}, \end{equation} with $K$ bounded on $\mathbb{R}$.

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arXiv2014

Tori in $S^3$ minimizing locally the conformal volume

We prove that the conformal immersions of complex two tori into $S^3$ which locally minimize their conformal volume in their conformal class all satisfy some elliptic PDE. We prove that they are either minimal tori, CMC flat tori, elliptic conformally constrained minimal tori or critical point of the area under some fixed conformally congruent area. On the way to establish this result we prove that tori which are critical points of the area for perturbations within a given conformal class and which are degenerate points of the conformal class mapping - i.e. isothermic - are either minimal surfaces or flat CMC tori. These results are all proved in the general framework of weak immersions.

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arXiv2014

Bourbaki Seminar 1081 : Min-max methods and the Willmore conjecture, after Fernando Codá Marques and André Arroja Neves

Two years ago, F.C. Marques and A.A. Neves implemented, in the framework of closed rectifiable 2-dimensional currents of the 3-dimensional sphere, a min-max method in geometric measure theory due to F. Almgren and J. Pitts. Using this approach they succeeded in proving that the famous Clifford torus minimizes the area among all closed minimal surfaces of non-zero genus in $S^3$. Another spectacular consequence of their work is to provide a proof of the Willmore conjecture. The goal of this talk is to discuss first the general framework of these two theorems of Marques and Neves. We shall then present the structures and some key details of their proofs. We will then address the scope of this remarkable contribution to the calculus of variations on surfaces in a 3-dimensional space.

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