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Tristan Rivière

Publications and source records attributed to Tristan Rivière.

At least 19 recordsLinked to original sources
arXiv2026

Improved Morse Index Stability for Sequences of Harmonic Maps from Degenerating Riemann Surfaces

We study the stability of the extended Morse index, defined as the number of negative and zero eigenvalues of the Jacobi operator, for sequences of harmonic maps on degenerating Riemann surfaces. As the conformal structure approaches the boundary of moduli space, collar collapse creates major analytical challenges. We analyze the second variation of the energy under these degenerations and identify conditions ensuring upper semicontinuity of the extended Morse index. Refining earlier results of the first and second authors in [7], we obtain sharper control of the spectrum of the Jacobi operator on degenerating domains. A key new aspect is the explicit contribution of geodesics arising as limits of the images of degenerating collars. We show that these neck regions converge to geodesic segments whose Morse index contributes nontrivially to the limiting extended index.

math.DG↗
arXiv2026

Gauge Symmetry Breaking in the Asymptotic Analysis of Self Dual Yang-Mills-Higgs $SU(2)$ Monopoles

We consider the $SU(2)$ Self-Dual Yang Mills Higgs Lagrangian in 3 dimension. By adding a ''Gauge Mass'' term to this YMH Lagrangian in the form of $L^2$ norm of the connection we break the gauge invariance and critical points are automatically fulfilling globally the Coulomb condition. We study the so called ``large mass asymptotic'', which has the effect of ''squeezing'' the monopoles. For any unit Higgs field data at the boundary we prove that minimizers of this Coulomb-Yang-Mills-Higgs Functional converge to harmonic maps into ${\mathbb S}^2$ extending this data. This asymptotic moreover is subject to concentration conpactness phenomena and the convergence is strong away from a 1 dimensional rectifiable closed concentration set. Then we prove that, having chosen a large enough coupling constant, the limiting minimal energy is converging towards the minimal Brezis-Coron-Lieb relaxed harmonic map energy for this boundary data. In the second part of the paper we examine a different asymptotic regime characterised by overloading monopoles. In this regime we prove that asymptotically, the magnetic field becomes exclusively longitudinal with a $U(1)$ abelian component along the Higgs Field while the Higgs field itself converges to a smooth absolute minimizer of a relaxation of the Faddeev-Skyrme functional of maps from ${\mathbb B}^3$ into ${\mathbb S}^2$. In the third part of the paper we study the behaviour of these configurations when the parameter in front of the Fadeev-Skyrme component respectively goes to zero and $+\infty$. In the first case one recovers the Brezis Coron Lieb relaxed energy at the limit while in the second case the minimal limiting energy is converging towards the minimal Dirichlet energy of maps into ${\mathbb S}^3$ whose projection by the Hopf fibration is equal to the fixed boundary data.

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arXiv2026

The Regularity of Critical Points to Scale-Invariant Curvature Energies in Dimension 4

We consider a class of scale-invariant curvature energies defined on immersed $4$-dimensional manifolds and prove that weak immersions that are critical points of such energies are analytic in any given local harmonic chart. Because of the criticality of this variational problem, the regularity result is obtained through the identification of conservation laws by applying Noether theorem. The resulting identities generate a lower order elliptic system of PDEs to which methods from integrability by compensation and interpolation theory are applied.

math.AP↗
arXiv2025

Morse index stability for p-Yang-Mills connections

We establish the lower semi continuity of the Morse index and the upper continuity of the Morse Index plus nullity of sequences of critical points of the Sacks-Uhlenbeck type relaxation of the Yang-Mills Energy in 4 dimension. The result is known not to be true in general for the ``cousin problem'' of hamonic maps from surfaces into arbitrary manifolds. This result is stressing the more stable behaviour of Yang-Mills Fields compare to harmonic maps as observed in other contexts such as the flow. The Morse Index control at the limit of critical points to Sacks Uhlenbeck relaxations of Yang-Mills Lagrangian is a central result in the implementation of minmax operation on this Lagrangian.

math.DG↗
arXiv2025

Weak immersions with second fundamental form in a critical Sobolev space

We develop the analysis of Lipschitz immersions of $n$-dimensional manifolds into $\mathbb{R}^d$ having their second fundamental forms bounded in the critical Sobolev space $W^{\frac{n}{2}-1,2}$ in dimension $n\geq 4$ even and any codimension. We prove that, while such a weak immersion is not necessary $C^1$, it generates a $C^1$ differential structure on the domain. More precisely, for any such an immersion, there is an atlas in which the first fundamental form is continuous and the transition maps are $C^1$. We prove that this $C^1$ structure is diffeomorphic to the original one. This result is the starting point of the analysis of the behavior of sequences of weak immersions with second fundamental forms uniformly bounded in the critical Sobolev space $W^{\frac{n}{2}-1,2}$. In the second part of the paper we establish a weakly sequential closure theorem for such sequences. This analysis is motivated by the study of conformally invariant Lagrangian of immersions in dimension larger than two such as generalized Willmore energies, for instance the Graham--Reichert functional obtained in the computation of renormalized volumes of five-dimensional minimal submanifolds of the hyperbolic space $\mathbb{H}^{d+1}$.

math.DG↗
arXiv2025

Construction of harmonic coordinates for weak immersions

We prove that any weak immersion in the critical Sobolev space $W^{\frac{n}{2}+1,2}(\mathbb{R}^n;\mathbb{R}^d)$ in even dimension $n\geq 4$, has global harmonic coordinates if its second fundamental form is small in the Sobolev space $W^{\frac{n}{2}-1,2}(\mathbb{R}^n;\mathbb{R}^d)$. This is a generalization to arbitrary even dimension $n\ge 4$ of a famous result of Müller--Sverak \cite{muller1995} for $n=2$. The existence of such coordinates is a key tool used by the authors in \cite{MarRiv20252} for the analysis of scale-invariant Lagrangians of immersions, such as the Graham--Reichert functional. From a purely intrinsic perspective, the proof of the main result leads to a general local existence theorem of harmonic coordinates for general metrics with Riemann tensor in $L^p$ for any $p>n/2$ in any dimension $n\geq 3$.

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arXiv2025

Gluing instantons à la Brezis-Coron in dimension four and the dipole construction

Given a connection $A$ on a $SU(2)$-bundle $P$ over $\mathbb{R}^4$ with finite Yang-Mills energy $YM(A)$ and nonzero curvature $F_A(0)$ at the origin, and given $ρ>0$ small enough, we construct a new connection $\hat A$ on a bundle $\hat P$ of different Chern class ($|c_2(A)-c_2(\hat A)|=8π^2$), in such a way that $\hat A$ is gauge equivalent to $A$ in $\mathbb{R}^4\setminus B_ρ(0)$, gauge equivalent to an instanton in a smaller ball $B_{τρ}(0)$, and $$YM(\hat A)\le YM(A)+8π^2-\varepsilon_0ρ^4|F_A(0)|^2,$$ where $τ\in (0.3,0.4)$ and $\varepsilon_0>0$ are universal constant independent of $A$ and $ρ$. Our gluing method is similar in spirit to the one of Brezis-Coron for harmonic maps. We compare it with classical results by Taubes and discuss applications and open problems.

math.DG↗
arXiv2025

Minimizing the Gauss map area of surfaces in $\mathbb{S}^3$

We establish the lower bound of $4π(1+g)$ for the area of the Gauss map of any immersion of a closed oriented surface of genus $g$ into $\mathbb{S}^3$, taking values in the Grassmannian of $2$-planes in $\mathbb{R}^4$. This lower bound is proved to be optimal for any genus $g \in \mathbb{N}$ but attained only when $g = 0$. For $g \neq 0$ we describe the behavior of any minimizing sequence of embeddings: we prove that, modulo extraction of a subsequence, the surfaces converge in the Hausdorff distance to a round sphere $S$, and the integral cycles carried by the Gauss maps split into $g+1$ spheres, each of area $4π$; one of them corresponds to the cycle carried by the Gauss map of $S$, while the other $g$ arise from the concentration of negative Gauss curvature at $g$ points of $S$. The results of this paper are used by the second author to define a nontrivial homological 4-dimensional min-max scheme for the area of Gauss maps of immersions into $\mathbb{S}^3$ in relation to the Willmore conjecture.

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arXiv2025

Coulomb Gauges and Regularity for Stationary Weak Yang$-$Mills Connections in Supercritical Dimension

We prove that stationary Yang$-$Mills fields in dimensions 5 belonging to the variational class of weak connections are smooth away from a closed singular set $S$ of vanishing 1-dimensional Hausdorff measure. Our proof is based on an $\varepsilon$-regularity theorem, which generalizes to this class of weak connections the existing previous $\varepsilon$-regularity results by G. Tian for smooth connections, by Y. Meyer and the second author for Sobolev and approximable connections, and by T. Tao and G. Tian for admissible connections (which are weak limits of smooth Yang$-$Mills fields). On the path towards establishing $\varepsilon$-regularity, a pivotal step is the construction of controlled Coulomb gauges for general weak connections under small Morrey norm assumptions.

math.DG↗
arXiv2025

Morse Index Stability for Sequences of Sacks-Uhlenbeck Maps into a Sphere

In this paper we consider sequences of $p$-harmonic maps, $p>2$, from a closed Riemann surface $Σ$ into the $n$-dimensional sphere $\mathbb{S}^n$ with uniform bounded energy. These are critical points of the energy $E_p(u) :=\int_Σ\left( 1+|{\nabla u}|^2\right)^{p/2} \ dvol_Σ.$ Our two main results are an improved pointwise estimate of the gradient in the neck regions around blow up points and the proof that the necks are asymptotically not contributing to the negativity of the second variation of the energy $E_p.$ This allows us, in the spirit of the paper of the first and second authors in collaboration with M. Gianocca {\em Morse index stability for critical points to conformally invariant Lagrangians}, to show the upper semicontinuity of the Morse index plus nullity for sequences of $p$-harmonic maps into a sphere.

math.AP↗
arXiv2024

A Variational Theory for The Area of Legendrian Surfaces

We study a new notion of critical point for the area of surfaces under the Legendrian constraint, called parametrized Hamiltonian stationary Legendrian varifolds (PHSLVs). We establish several fundamental properties of these objects, including their sequential compactness and an optimal regularity result, showing that they are smooth immersions away from a locally finite set of branch points and Schoen Wolfson conical singularities. This generalizes in particular the regularity theory of Schoen Wolfson for minimizers to general critical points. This theory can be used to show two new variational results: every minmax operation with the area of (closed, immersed) Legendrian surfaces in a closed Sasakian 5-dimensional manifold is achieved by a Hamiltonian stationary map with this regularity; also, the minimal area in any given exact isotopy class of Legendrian immersions of $S^2$ is realized by such a map. Along the way, we prove an effective monotonicity formula for general two-dimensional stationary varifolds in the Legendrian setting, as well as the closure of integral stationary varifolds among rectifiable ones, in spite of the lack of compactness of the latter.

math.DG↗
arXiv2024

Area Variations under Legendrian Constraint

In any 5 dimensional closed Sasakian manifold we prove that any minmax operation on the area among Legendrian surfaces is achieved by a continuous conformal Legendrian map from a closed riemann surface $S$ into $N^5$ equipped with an integer multiplicity bounded in $L^\infty$. Moreover this map, equipped with this multiplicity, satisfies a weak version of the Hamiltonian Minimal Equation. We conjecture that any solution to this equation is a smooth branched Legendrian immersion away from isolated Schoen-Wolfson conical singularities with non zero Maslov class.

math.DG↗
arXiv2023

A Variational Construction of Hamiltonian Stationary Surfaces with Isolated Schoen-Wolfson Conical Singularities

We construct using variational methods Hamiltonian Stationary Surfaces with Isolated Schoen-Wolfson Conical Singularities. We obtain these surfaces through a convergence process reminiscent to the Ginzburg-Landau asymptotic analysis in the strongly repulsive regime. We describe in particular how the prescription of Schoen Wolfson conical singularities is related to optimal Wente constants.

math.DG↗
arXiv2023

A variational approach to $S^1$-harmonic maps and applications

We present a renormalization procedure of the Dirichlet Lagrangian for maps from surfaces with or without boundary into $S^1$ and whose finite energy critical points are the $S^1-$harmonic maps with isolated singularities. We give some applications of this renormalization scheme in two different frameworks. The first application has to do with the renormalization of the Willmore energy for Lagrangian singular immersions into Kähler-Einstein Surfaces while the second application is dealing with frame energies for surfaces immersions into Euclidian spaces.

math.DG↗
arXiv2023

The Parametric Willmore Flow

We establish a minimal positive existence time of the parametric Willmore flow for any smooth initial data (smooth immersion of a closed oriented surface). The minimal existence time is a function exclusively of geometric data which in particular are all well defined for general weak lipschitz $W^{2,2}$ immersions. This fact combined with the conservation law formulation of the equation given by the first author in 2008 opens the possibility for defining the Willmore flow for weak lipschitz $W^{2,2}$ initial data.

math.DG↗
arXiv2023

Morse Index Stability of Willmore Immersions I

In a recent work, F. Da Lio, M. Gianocca, and T. Rivière 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. Rivière 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.

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