Robustness of Liouville Measure under a Family of Stable Diffusions

Consider a C^∞ closed connected Riemannian manifold (M, g) with negative sectional curvature. The unit tangent bundle SM is foliated by the (weak) stable foliation of the geodesic flow. Let Δ^s be the leafwise Laplacian for and let X be the geodesic spray, i.e., the vector field that generates the geodesic flow. For each ρ , the operator generates a diffusion for . We show that, as ρ → − ∞ , the unique stationary probability measure for the leafwise diffusion of converge to the normalized Liouville measure on SM . © 2020 Wiley Periodicals LLC     

Unique Continuation at the Boundary for Harmonic Functions in C1 Domains and Lipschitz Domains with Small Constant

Let be a domain, or more generally, a Lipschitz domain with small local Lipschitz constant. In this paper it is shown that if u is a function harmonic in and continuous in , which vanishes in a relatively open subset ; moreover, the normal derivative vanishes in a subset of with positive surface measure; then u is identically zero. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

On Heavy-Tail Phenomena in Some Large-Deviations Problems

We revisit the proof of the large-deviations principle of Wiener chaoses partially given by Borell and then by Ledoux in its full form. We show that some heavy-tail phenomena observed in large deviations can be explained by the same mechanism as for the Wiener chaoses, meaning that the deviations are created, in a sense, by translations. More precisely, we prove a general large-deviations principle for a certain class of functionals , where is some metric space, under the n -fold probability measure , where α ∈ (0, 2] , for which the large deviations are due to translations. We retrieve, as an application, the large-deviations principles known for the Wigner matrices without Gaussian tails, in works by Bordenave and Caputo on one hand, and the author on the other hand, of the empirical spectral measure, the largest eigenvalue, and traces of polynomials. We also apply our large-deviations result to the last-passage time, which yields a large-deviations principle when the weights follow the law , with α ∈ (0, 1) . © 2020 Wiley Periodicals LLC     

From 1 to 6: A Finer Analysis of Perturbed Branching Brownian Motion

The logarithmic correction for the order of the maximum for two-speed branching Brownian motion changes discontinuously when approaching slopes , which corresponds to standard branching Brownian motion. In this article we study this transition more closely by choosing and . We show that the logarithmic correction for the order of the maximum now smoothly interpolates between the correction in the i.i.d. case , and when . This is due to the localization of extremal particles at the time of speed change, which depends on α and differs from the one in standard branching Brownian motion. We also establish in all cases the asymptotic law of the maximum and characterize the extremal process, which turns out to coincide essentially with that of standard branching Brownian motion. © 2020 the Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC     

Normalized Harmonic Map Heat Flow

For any closed Riemannian manifold N we propose the normalized harmonic map heat flow as a means to obtain nonconstant harmonic maps , m ≥ 3 . © 2019 Wiley Periodicals, Inc.     

A Liouville Theorem for the Complex Monge-Ampère Equation on Product Manifolds

Let Y be a closed Calabi-Yau manifold. Let ω be the Kähler form of a Ricci-flat Kähler metric on . We prove that if ω is uniformly bounded above and below by constant multiples of , where is the standard flat Kähler form on and ω_Y is any Kähler form on Y, then ω is a product Kähler form up to a certain automorphism of . © 2018 Wiley Periodicals, Inc.     

Inviscid Limit of Vorticity Distributions in the Yudovich Class

We prove that given initial data , forcing and any T > 0, the solutions u^ν of Navier-Stokes converge strongly in for any p ∈ [1, ∞) to the unique Yudovich weak solution u of the Euler equations. A consequence is that vorticity distribution functions converge to their inviscid counterparts. As a by-product of the proof, we establish continuity of the Euler solution map for Yudovich solutions in the L^p vorticity topology. The main tool in these proofs is a uniformly controlled loss of regularity property of the linear transport by Yudovich solutions. Our results provide a partial foundation for the Miller-Robert statistical equilibrium theory of vortices as it applies to slightly viscous fluids. © 2020 Wiley Periodicals LLC.     

Upper Tail Large Deviations of Regular Subgraph Counts in Erdős-Rényi Graphs in the Full Localized Regime

For a -regular connected graph H the problem of determining the upper tail large deviation for the number of copies of H in , an Erdős-Rényi graph on n vertices with edge probability p, has generated significant interest. For and , where is the number of vertices in H, the upper tail large deviation event is believed to occur due to the presence of localized structures. In this regime the large deviation event that the number of copies of H in exceeds its expectation by a constant factor is predicted to hold at a speed , and the rate function is conjectured to be given by the solution of a mean-field variational problem. After a series of developments in recent years, covering progressively broader ranges of p, the upper tail large deviations for cliques of fixed size were proved by Harel, Mousset, and Samotij in the entire localized regime. This paper establishes the conjecture for all connected regular graphs in the whole localized regime. © 2021 Wiley Periodicals LLC.     

Sharp Estimate of Global Coulomb Gauge

Let A be a W^{1, 2} -connection on a principal SU(2) -bundle P over a compact 4 -manifold M whose curvature F_A satisfies . Our main result is the existence of a global section σ : M → P with finite singularities on M such that the connection form σ^*A satisfies the Coulomb equation d^*(σ^*A) = 0 and admits a sharp estimate . Here ℒ^{4, ∞} is a new function space we introduce in this paper that satisfies L⁴(M) ⊊ ℒ^{4, ∞}(M) ⊊ L^{4 − ϵ}(M) for all ϵ > 0 . More precisely, ℒ^{4, ∞}(M) is the collection of measurable function u such that , where L^{4, ∞} is the classical Lorentz space and s_u is the L⁴ -integrability radius function associated to u defined by Briefly speaking, we achieve the estimate of by showing that σ^*A is effectively L⁴ -integrable away from controllably many points on M . © 2020 Wiley Periodicals LLC     

Improved Moser-Trudinger-Onofri Inequality under Constraints

A classical result of Aubin states that the constant in the Moser-Trudinger-Onofri inequality on can be improved for functions with zero first-order moments of the area element. We generalize it to the higher-order moments case. These new inequalities bear similarity to a sequence of Lebedev-Milin-type inequalities on coming from the work of Grenander-Szego on Toeplitz determinants (as pointed out by Widom). We also discuss the related sharp inequality by a perturbation method . © 2020 Wiley Periodicals LLC.     

Geometry and Temperature Chaos in Mixed Spherical Spin Glasses at Low Temperature: The Perturbative Regime

We study the Gibbs measure of mixed spherical p -spin glass models at low temperature, in (part of) the 1-RSB regime, including, in particular, models close to pure in an appropriate sense. We show that the Gibbs measure concentrates on spherical bands around deep critical points of the (extended) Hamiltonian restricted to the sphere of radius , where is the rightmost point in the support of the overlap distribution. We also show that the relevant critical points are pairwise orthogonal for two different low temperatures. This allows us to explain why temperature chaos occurs for those models, in contrast to the pure spherical models. © 2019 Wiley Periodicals, Inc.     

Global Regularity of 2D Density Patches for Viscous Inhomogeneous Incompressible Flow with General Density: Low Regularity Case

This paper presents some progress toward an open question proposed by P.-L. Lions [26] concerning the propagation of regularities of density patches for viscous inhomogeneous incompressible flow. We first establish the global-in-time well-posedness of the two-dimensional inhomogeneous incompressible Navier-Stokes system with initial density . Here is any pair of positive constants and Ω₀ is a bounded, simply connected domain. We then prove that for any positive time t, the density , with the domain Ω(t) preserving the -boundary regularity. © 2018 Wiley Periodicals, Inc.     

Fully Nonlinear Equations with Applications to Grad Equations in Plasma Physics

In this paper we generalize an equation studied by Mossino and Temam in [7], to the fully nonlinear case. This equation arises in plasma physics as an approximation to Grad equations, which were introduced by Harold Grad in [4], to model the behavior of plasma confined in a toroidal vessel called TOKAMAK. We prove existence of a -viscosity solution and regularity up to for any (we improve this regularity near the boundary). The difficulty of this problem lies in the right-hand side which involves the measure of the superlevel sets, making the problem nonlocal. © 2021 Wiley Periodicals LLC.     

Identification of the Polaron Measure I: Fixed Coupling Regime and the Central Limit Theorem for Large Times

We consider the Fröhlich model of the polaron, whose path integral formulation leads to the transformed path measure with respect to ℙ that governs the law of the increments of the three-dimensional Brownian motion on a finite interval [−T, T] , and Z_{α, T} is the partition function or the normalizing constant and α > 0 is a constant, or the coupling parameter. The polaron measure reflects a self-attractive interaction. According to a conjecture of Pekar that was proved in [9], exists and has a variational formula. In this article we show that when α > 0 is either sufficiently small or sufficiently large, the limit exists, which is also identified explicitly. As a corollary, we deduce the central limit theorem for under and obtain an expression for the limiting variance. © 2019 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, LLC.     

Min-Oo Conjecture for Fully Nonlinear Conformally Invariant Equations

In this paper we show rigidity results for supersolutions to fully nonlinear, elliptic, conformally invariant equations on subdomains of the standard n -sphere under suitable conditions along the boundary. We emphasize that our results do not assume concavity on the fully nonlinear equations we will work with. This proves rigidity for compact, connected, locally conformally flat manifolds (M, g) with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality and whose boundary is isometric to a geodesic sphere ∂D(r) , where D(r) denotes a geodesic ball of radius r ∈ (0, π/2] in , and totally umbilical with mean curvature bounded below by the mean curvature of this geodesic sphere. Under the above conditions, (M, g) must be isometric to the closed geodesic ball . As a side product, in dimension 2 our methods provide a new proof to Toponogov's theorem about rigidity of compact surfaces carrying a shortest simple geodesic. Roughly speaking, Toponogov's theorem is equivalent to a rigidity theorem for spherical caps in the hyperbolic three-space ℍ³ . In fact, we extend it to obtain rigidity for supersolutions to certain Monge-Ampère equations. © 2019 Wiley Periodicals, Inc.     

Asymptotic Stability of Harmonic Maps on the Hyperbolic Plane under the Schrödinger Maps Evolution

We consider the Cauchy problem for the Schrödinger maps evolution when the domain is the hyperbolic plane. An interesting feature of this problem compared to the more widely studied case on the Euclidean plane is the existence of a rich new family of finite energy harmonic maps. These are stationary solutions, and thus play an important role in the dynamics of Schrödinger maps. The main result of this article is the asymptotic stability of (some of) such harmonic maps under the Schrödinger maps evolution. More precisely, we prove the nonlinear asymptotic stability of a finite energy equivariant harmonic map under the Schrödinger maps evolution with respect to non-equivariant perturbations, provided obeys a suitable linearized stability condition. This condition is known to hold for all equivariant harmonic maps with values in the hyperbolic plane and for a subset of those maps taking values in the sphere. One of the main technical ingredients in the paper is a global-in-time local smoothing and Strichartz estimate for the operator obtained by linearization around a harmonic map, proved in the companion paper [36]. © 2021 Wiley Periodicals LLC.     

Upper Bounds on Liouville First-Passage Percolation and Watabiki's Prediction

Given a planar continuum Gaussian free field h^𝒰 in a domain 𝒰 with Dirichlet boundary condition and any δ > 0, we let be a real-valued smooth Gaussian process where is the average of h^𝒰 along a circle of radius δ with center v. For γ > 0, we study the Liouville first-passage percolation (in scale δ), i.e., the shortest path distance in 𝒰 where the weight of each path P is given by . We show that the distance between two typical points is for all sufficiently small but fixed γ > 0 and some constant c^* > 0. In addition, we obtain similar upper bounds on the Liouville first-passage percolation for discrete Gaussian free fields, as well as the Liouville graph distance, which roughly speaking is the minimal number of euclidean balls with comparable Liouville quantum gravity measure whose union contains a continuous path between two endpoints. Our results contradict some reasonable interpretations of Watabiki's prediction (1993) on the random distance of Liouville quantum gravity at high temperatures.© 2019 Wiley Periodicals, Inc.     

A De Giorgi–Type Conjecture for Minimal Solutions to a Nonlinear Stokes Equation

We study the one-dimensional symmetry of solutions to the nonlinear Stokes equation which are periodic in the d − 1 last variables (living on the torus 𝕋^d−1) and globally minimize the corresponding energy in Ω = ℝ × 𝕋^d−1, i.e., Namely, we find a class of nonlinear potentials W ≥ 0 such that any global minimizer u of E connecting two zeros of W as x₁ → ± ∞ is one-dimensional; i.e., u depends only on the x₁ -variable. In particular, this class includes in dimension d = 2 the nonlinearities with w being a harmonic function or a solution to the wave equation, while in dimension d ≥ 3 , this class contains a perturbation of the Ginzburg-Landau potential as well as potentials W having d + 1 wells with prescribed transition cost between the wells. For that, we develop a theory of calibrations relying on the notion of entropy (coming from scalar conservation laws). We also study the problem of the existence of global minimizers of E for general potentials W providing in particular compactness results for uniformly finite energy maps u in Ω connecting two wells of W as x₁ → ± ∞ . © 2019 Wiley Periodicals, Inc.