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Paul Laurain

Publications and source records attributed to Paul Laurain.

At least 19 recordsLinked to original sources
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.

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arXiv2025

Huber Theorem revisited in dimensions 2 and 4

We study the second Huber theorem in dimensions 2 and 4. In dimension 2, we prove a new version assuming that the Gauss curvature lies in a negative Sobolev space using Coulomb frames. In dimension $4$, given a metric having a pointwise singularity with $L^p$-bounds on the Bach tensor, we construct a conformal metric which is regular across the singularity. To do so, we introduce another Coulomb-type condition, similar to the case of Yang--Mills connections. This enables us to obtain a conformal metric satisfying an $\varepsilon$-regularity property. We obtain a generalization of the two-dimensional case that can be applied to study the singularities of Bach-flat metrics and immersions with second fundamental forms in $W^{2,\frac{4}{3}+\varepsilon}$.

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arXiv2024

Morse index stability for Yang-Mills connections

We prove stability results of the Morse index plus nullity of Yang-Mills connections in dimension 4 under weak convergence. Precisely we establish that the sum of the Morse indices and the nullity of a bounded sequence of Yang-Mills connections is asymptotically bounded above by the sum of the Morse index and the nullity of the weak limit and the bubbles while the Morse indices are asymptotically bounded below by the sum of the Morse index of the weak limit and the bubbles.

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arXiv2022

Stability of the Pohožaev obstrucion in dimension 3

We investigate problems connected to the stability of the wellknown Pohožaev obstruction. We generalize results which were obtained in the minimizing setting by Brezis and Nirenberg [2] and more recently in the radial situation by Brezis and Willem [3].

math.AP↗
arXiv2022

An Obstruction to the Existence of Immersed Curves of Prescribed Curvature

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arXiv2022

Fine multibubble analysis in the higher-dimensional Brezis-Nirenberg problem

For a bounded set $\Omega \subset \mathbb R^N$ and a perturbation $V \in C^1(\overline{\Omega})$, we analyze the concentration behavior of a blow-up sequence of positive solutions to \[ -\Delta u_\epsilon + \epsilon V = N(N-2) u_\epsilon^\frac{N+2}{N-2} \] for dimensions $N \geq 4$, which are non-critical in the sense of the Brezis--Nirenberg problem. For the general case of multiple concentration points, we prove that concentration points are isolated and characterize the vector of these points as a critical point of a suitable function derived from the Green's function of $-\Delta$ on $\Omega$. Moreover, we give the leading order expression of the concentration speed. This paper, with a recent one by the authors (arXiv:2208.12337) in dimension $N = 3$, gives a complete picture of blow-up phenomena in the Brezis-Nirenberg framework.

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arXiv2022

Multibubble blow-up analysis for the Brezis-Nirenberg problem in three dimensions

For a smooth bounded domain $\Omega \subset \mathbb R^3$ and smooth functions $a$ and $V$, we consider the asymptotic behavior of a sequence of positive solutions $u_\epsilon$ to $-\Delta u_\epsilon + (a+\epsilon V) u_\epsilon = u_\epsilon^5$ on $\Omega$ with zero Dirichlet boundary conditions, which blow up as $\epsilon \to 0$. We derive the sharp blow-up rate and characterize the location of concentration points in the general case of multiple blow-up, thereby obtaining a complete picture of blow-up phenomena in the framework of the Brezis-Peletier conjecture in dimension $N=3$.

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arXiv2022

Rigidity Theorems for Asymptotically Euclidean $Q$-singular Spaces

In this paper we prove some rigidity theorems associated to $Q$-curvature analysis on asymptotically Euclidean (AE) manifolds, which are inspired by the analysis of conservation principles within fourth order gravitational theories. A central object in this analysis is a notion of fourth order energy, previously analysed by the authors, which is subject to a positive energy theorem. We show that this energy can be more geometrically rewritten in terms of a fourth order analogue to the Ricci tensor, which we denote by $J_g$. This allows us to prove that Yamabe positive $J$-flat AE manifolds must be isometric to Euclidean space. As a by product, we prove that this $J$-tensor provides a geometric control for the optimal decay rates at infinity. This last result reinforces the analogy of $J$ as a fourth order analogue to the Ricci tensor.

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arXiv2022

Constant $Q$-curvature metrics with a singularity

For dimensions $n \geq 3$, we classify singular solutions to the generalized Liouville equation $(-Δ)^{n/2} u = e^{nu}$ on $\mathbb{R}^n \setminus \{0\}$ with the finite integral condition $\int_{\mathbb{R}^n} e^{nu} < \infty$ in terms of their behavior at $0$ and $\infty$. These solutions correspond to metrics of constant $Q$-curvature which are singular in the origin. Conversely, we give an optimal existence result for radial solutions. This extends some recent results on solutions with singularities of logarithmic type to allow for singularities of arbitrary order. As a key tool to the existence result, we derive a new weighted Moser--Trudinger inequality for radial functions.

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arXiv2021

On the Positive Energy Theorem for Stationary Solutions to Fourth-Order Gravity

In this paper we prove a positive energy theorem related to fourth-order gravitational theories, which is a higher-order analogue of the classical ADM positive energy theorem of general relativity. We will also show that, in parallel to the corresponding situation in general relativity, this result intersects several important problems in geometric analysis. For instance, it underlies positive mass theorems associated to the Paneitz operator, playing a similar role in the positive $Q$-curvature conformal prescription problem as the Schoen-Yau positive energy theorem does for the Yamabe problem. Several other links to $Q$-curvature analysis and rigidity phenomena are established.

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arXiv2020

Energy Estimates for the Tracefree Curvature of Willmore Surfaces and Applications

We prove an $ε$-regularity result for the tracefree curvature of a Willmore surface with bounded second fundamental form. For such a surface, we obtain a pointwise control of the tracefree second fundamental form from a small control of its $L^2$-norm.Several applications are investigated. Notably, we derive a gap statement for surfaces of the aforementioned type. We further apply our results to deduce regularity results for conformal minimal spacelike immersions into the de Sitter space $S^{4,1}$.

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arXiv2019

Classification of uniformly distributed measures of dimension $1$ in general codimension

Starting with the work of Preiss on the geometry of measures, the classification of uniform measures in $\mathbb R^d$ has remained open, except for $d=1$ and for compactly supported measures in $d=2$, and for codimension $1$. In this paper we study $1$-dimensional measures in $\mathbb R^d$ for all $d$ and classify uniform measures with connected $1$-dimensional support, which turn out to be homogeneous measures. We provide as well a partial classification of general uniform measures of dimension $1$ in the absence of the connected support hypothesis.

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arXiv2018

Existence of Min-Max Free Boundary Disks Releasing the Width of a Manifold

We perform a replacement procedure in order to produce a free boundary minimal surface whose area achieves the min-max value over all disk sweepouts of a manifold whose boundary lie in a submanifold. Our result is based on a proof of the convexity of the energy for free boundary harmonic maps and a generalization of Colding-Minicozzi replacement procedure.

math.DG↗
arXiv2018

Energy convexity of intrinsic bi-harmonic maps and applications I: spherical target

Every harmonic map is an intrinsic bi-harmonic map as an absolute minimizer of the intrinsic bi-energy functional, therefore intrinsic bi-harmonic map and its heat flow are more geometrically natural to study, but they are also considerably more difficult analytically than the extrinsic counterparts due to the lack of coercivity for the intrinsic bi-energy. In this paper, we show an energy convexity and thus uniqueness for weakly intrinsic bi-harmonic maps from the unit $4$-ball $B_1 \subset \mathbf{R}^4$ into the sphere $\mathbf{S}^n$. This is a higher-order analogue of the energy convexity and uniqueness for weakly harmonic maps on unit $2$-disk in $\mathbf{R}^2$ proved by Colding and Minicozzi \cite{CM08} (see also Lamm and the second author \cite{LL13}). In particular, this yields a version of uniqueness of weakly harmonic maps on the unit $4$-ball which is new. As an application, we also show a version of energy convexity along the intrinsic bi-harmonic map heat flow into $\mathbf{S}^n$, which in particular yields the long-time existence of the intrinsic bi-harmonic map heat flow, a result that was until now only known assuming the non-positivity of the target manifolds by Lamm \cite{Lamm05}. Moreover, the energy convexity along the flow yields the uniform convergence of the flow which is not known before. One of the key ingredients in our proofs is a refined version of the $ε$-regularity of the first author and Rivière \cite{LaR}.

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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 $12π$.

math.DG↗
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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