We consider compact ancient solutions to the three‐dimensional Ricci flow that are κ‐noncollapsed. We prove that such a solution either is a family of shrinking round spheres or has a unique asymptotic behavior as t → − ∞, which we describe. This analysis applies in particular to the ancient solution constructed by Perelman. © 2020 Wiley Periodicals LLC

Let A be a W1, 2‐connection on a principal SU(2)‐bundle P over a compact 4‐manifold M whose curvature FA 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 ℒ4, ∞ is a new function space we introduce in this paper that satisfies L4(M) ⊊ ℒ4

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 Δs 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

Soap films at equilibrium are modeled, rather than as surfaces, as regions of small total volume through the introduction of a capillarity problem with a homotopic spanning condition. This point of view introduces a length scale in the classical Plateau's problem, which is in turn recovered in the vanishing volume limit. This approximation of area minimizing hypersurfaces leads to an energy based selection

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

This work is concerned with an asymptotic analysis, in the sense of Γ‐convergence, of a sequence of variational models of brittle damage in the context of linearized elasticity. The study is performed as the damaged zone concentrates into a set of zero volume and, at the same time and to the same order ε, the stiffness of the damaged material becomes small. Three main features make the analysis highly

We deal with a parabolic‐elliptic chemotaxis system. It is known that finite‐time blowup occurs for a large class of initial data. However, there have been no results on exact blowup rate or detailed blowup behavior except a special radial solution given just formally in [13] and rigorously in [19, 10, 9]. Our aim is to show that for all radial blowup solutions, their blowup rate, and blowup profile

We consider a nonlocal aggregation equation with degenerate diffusion, which describes the mean‐field limit of interacting particles driven by nonlocal interactions and localized repulsion. When the interaction potential is attractive, it is previously known that all steady states must be radially decreasing up to a translation, but uniqueness (for a given mass) within the radial class was open, except

This paper is devoted to the asymptotic analysis of a thin film equation that describes the evolution of a thin liquid droplet on a solid support driven by capillary forces. We propose an analytic framework to rigorously investigate the connection between this model and Tanner's law, which claims: the edge velocity of a spreading thin film on a prewetted solid is approximately proportional to the cube

We prove a sharp logarithmic Sobolev inequality that holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Like the Michael‐Simon Sobolev inequality, this inequality includes a term involving the mean curvature. © 2020 Wiley Periodicals LLC

Using toric geometry we give an explicit construction of the compact steady solitons for pluriclosed flow first constructed in by the first author in 2019. This construction also reveals that these solitons are generalized Kähler in two distinct ways, with vanishing and nonvanishing Poisson structure. This gives the first examples of generalized Kähler structures with nonvanishing Poisson structure

We consider the Fröhlich polaron model in the strong coupling limit. It is well‐known that to leading order the ground state energy is given by the (classical) Pekar energy. In this work, we establish the subleading correction, describing quantum fluctuation about the classical limit. Our proof applies to a model of a confined polaron, where both the electron and the polarization field are restricted

In this paper, we study the transition threshold of the 3D Couette flow in Sobolev space at high Reynolds number Re. It was proved that if the initial velocity v0 satisfies for some c0 > 0 independent of Re, then the solution of the 3D Navier‐Stokes equations is global in time and does not transition away from the Couette flow. This result confirms the transition threshold conjecture proposed by Trefethen

For M a simple surface, the nonlinear statistical inverse problem of recovering a matrix field from discrete, noisy measurements of the SO(n)‐valued scattering data CΦ of a solution of a matrix ODE is considered (n ≥ 2). Injectivity of the map Φ ↦ CΦ was established by Paternain, Salo, and Uhlmann in 2012. A statistical algorithm for the solution of this inverse problem based on Gaussian process priors

We provide the first nontrivial upper bound for the chemical distance exponent in two‐dimensional critical percolation. Specifically, we prove that the expected length of the shortest horizontal crossing path of a box of side length n in critical percolation on ℤ2 is bounded by Cn2 − δπ3(n) for some δ > 0, where π3(n) is the “three‐arm probability to distance n.” This implies that the ratio of this

We prove that given initial data , forcing and any T > 0, the solutions uν 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

We compute the topological simple structure set of closed manifolds that occur as total spaces of flat bundles over lens spaces Sl/(ℤ/p) with fiber Tn for an odd prime p and l ≥ 3 provided that the induced ℤ/p‐action on π1(Tn) = ℤn is free outside the origin. To the best of our knowledge this is the first computation of the structure set of a topological manifold whose fundamental group is not obtained

We study asymptotic behaviors of positive solutions to the Yamabe equation and the σk‐Yamabe equation near isolated singular points and establish expansions up to arbitrary orders. Such results generalize an earlier pioneering work by Caffarelli, Gidas, and Spruck and a work by Korevaar, Mazzeo, Pacard, and Schoen on the Yamabe equation and a work by Han, Li, and Teixeira on the σk‐Yamabe equation

The detailed behavior of solutions to Stokes equations on regions with corners has been historically difficult to characterize. The solutions to Stokes equations on regions with corners are known to develop singularities in the vicinity of corners; in particular, the solutions are known to have infinite oscillations along almost every ray that meet at the corner. While the nature of singularities for

Despite great progress in the study of critical percolation on ℤd for d large, properties of critical clusters in high‐dimensional fractional spaces and boxes remain poorly understood, unlike the situation in two dimensions. Closely related models such as critical branching random walk give natural conjectures for the value of the relevant high‐dimensional critical exponents; see in particular the

We prove the Yau‐Tian‐Donaldson conjecture for any ℚ‐Fano variety that has a log smooth resolution of singularities such that a negative linear combination of exceptional divisors is relatively ample and the discrepancies of all exceptional divisors are nonpositive. In other words, if such a Fano variety is K‐polystable, then it admits a Kähler‐Einstein metric. This extends the previous result for

We consider a driven tagged particle in a symmetric exclusion process on ℤ with a removal rule. In this process, untagged particles are removed once they jump to the left of the tagged particle. We investigate the behavior of the displacement of the tagged particle and prove limit theorems of it: an (annealed) law of large numbers, a central limit theorem, and a large deviation principle. We also characterize

By constructing an infinite‐dimensional KAM theorem of the normal frequencies being dense at a finite point, we show that some shallow water equations such as the Benjamin‐Bona‐Mahony equation and the generalized d ‐dimensional Pochhammer‐Chree equation subject to some boundary conditions possess many (a family of initial values of positive Lebesgue measure of finite dimension) smooth solutions that

The main result of this paper gives a new construction of extremal Kähler metrics on the total space of certain holomorphic submersions, giving a vast generalisation and unification of results of Hong, Fine and others. The principal new ingredient is a novel geometric partial differential equation on such fibrations, which we call the optimal symplectic connection equation.

It is well‐known that when the geometry and/or coefficients allow stable trapped rays, the outgoing solution operator of the Helmholtz equation grows exponentially through a sequence of real frequencies tending to infinity.

We prove an a priori bound for solutions of the dynamic equation. This bound provides a control on solutions on a compact space‐time set only in terms of the realisation of the noise on an enlargement of this set, and it does not depend on any choice of space‐time boundary conditions.

We prove a sharp transition for the infinitesimal distribution of a periodically banded GUE matrix. For bandwidths , we further prove that our model is infinitesimally free from the matrix units and the normalized all‐1’s matrix. Our results allow us to extend previous work of Shlyakhtenko on finite‐rank perturbations of Wigner matrices in the infinitesimal framework. For finite‐rank perturbations

Exterior problems for the maximal surface equation are studied. We obtain the precise asymptotic behavior of the exterior solution at infinity. We also prove that the exterior Dirichlet problem is uniquely solvable for admissible boundary data and prescribed asymptotic behavior at infinity. © 2020 Wiley Periodicals LLC

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 derive a multiscale generalisation of the Bakry‐Émery criterion for a measure to satisfy a log‐Sobolev inequality. Our criterion relies on the control of an associated PDE well‐known in renormalisation theory: the Polchinski equation. It implies the usual Bakry‐Émery criterion, but we show that it remains effective for measures that are far from log‐concave. Indeed, using our criterion, we prove

Let K be a convex polyhedron and ℱ its Wulff energy, and let denote the set of convex polyhedra close to K whose faces are parallel to those of K . We show that, for sufficiently small ε , all ε ‐minimizers belong to

Following van der Poorten, we consider a family of nonlinear maps that are generated from the continued fraction expansion of a function on a hyperelliptic curve of genus g. Using the connection with the classical theory of J ‐fractions and orthogonal polynomials, we show that in the simplest case g = 1 this provides a straightforward derivation of Hankel determinant formulae for the terms of a general

Many real‐world processes and phenomena are modeled using systems of ordinary differential equations with parameters. Given such a system, we say that a parameter is globally identifiable if it can be uniquely recovered from input and output data. The main contribution of this paper is to provide theory, an algorithm, and software for deciding global identifiability. First, we rigorously derive an

We focus on spherical spin glasses whose Parisi distribution has support of the form [0, q ]. For such models we construct paths from the origin to the sphere that consistently remain close to the ground‐state energy on the sphere of corresponding radius. The construction uses a greedy strategy, which always follows a direction corresponding to the most negative eigenvalues of the Hessian of the Hamiltonian

This paper proves the existence of small‐amplitude global‐in‐time unique mild solutions to both the Landau equation including the Coulomb potential and the Boltzmann equation without angular cutoff. Since the well‐known works [45] and [3, 43] on the construction of classical solutions in smooth Sobolev spaces which in particular are regular in the spatial variables, it still remains an open problem

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

We show in full generality the stability of optimal transport paths in branched transport: namely, we prove that any limit of optimal transport paths is optimal as well. This solves an open problem in the field (cf. Open problem 1 in the book Optimal transportation networks by Bernot, Caselles, and Morel), which has been addressed up to now only under restrictive assumptions. © 2020 Wiley Periodicals

We present a new proof of the sphere covering inequality in the spirit of comparison geometry, and as a by‐product we find another sphere covering inequality that can be viewed as the dual of the original one. We also prove sphere covering inequalities on surfaces satisfying general isoperimetric inequalities, and discuss their applications to elliptic equations with exponential nonlinearities in dimension

Based on the a priori estimates in our previous work, we continue to investigate the water‐waves problem in a bounded two‐dimensional corner domain in this paper. We prove the local well‐posedness of the solution to the water‐waves system when the contact angles are less than π /16. © 2020 Wiley Periodicals LLC

We study noncollapsed Gromov‐Hausdorff limits of Kähler manifolds with Ricci curvature bounded below. Our main result is that each tangent cone is homeomorphic to a normal affine variety. This extends a result of Donaldson‐Sun, who considered noncollapsed limits of polarized Kähler manifolds with two‐sided Ricci curvature bounds. © 2019 Wiley Periodicals LLC

Let PN be a uniform random N × N permutation matrix and let χN(z) = det(zIN − PN) denote its characteristic polynomial. We prove a law of large numbers for the maximum modulus of χN on the unit circle, specifically, with probability tending to 1 as N → ∞, for a numerical constant x0 ≈ 0.652. The main idea of the proof is to uncover a logarithmic correlation structure for the distribution of (the logarithm

We consider nonlinear, uniformly elliptic equations with random, highly oscillating coefficients satisfying a finite range of dependence. We prove that homogenization and linearization commute in the sense that the linearized equation (linearized around an arbitrary solution) homogenizes to the linearization of the homogenized equation (linearized around the corresponding solution of the homogenized

Analytic functions in the Hardy class H2 over the upper half‐plane ℍ+ are uniquely determined by their values on any curve Γ lying in the interior or on the boundary of ℍ+. The goal of this paper is to provide a sharp quantitative version of this statement. We answer the following question: Given f of a unit H2‐norm that is small on Γ (say, its L2‐norm is of order ϵ), how large can f be at a point

We consider a 2D vorticity configuration where vorticity is highly concentrated around a curve and exponentially decaying away from it: the intensity of the vorticity is O(1/ε) on the curve while it decays on an O(ε) distance from the curve itself. We prove that, if the initial datum is of vortex‐layer type, Euler solutions preserve this structure for a time that does not depend on ε. Moreover, the

In this paper we prove that the Benjamin‐Ono equation, when considered on the torus, is an integrable (pseudo)differential equation in the strongest possible sense: this equation admits global Birkhoff coordinates on the space of real‐valued, 2π‐periodic, L2‐integrable functions of mean 0. These are coordinates that allow us to integrate it by quadrature and hence are also referred to as nonlinear

The triple‐deck model is a classical high‐order boundary layer model that has been proposed to describe flow regimes where the Prandtl theory is expected to fail. At first sight the model appears to lose two derivatives through the pressure‐displacement relation that links pressure to the tangential slip. In order to overcome this, we split the triple‐deck system into two coupled equations: a Prandtl‐type

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

We provide a detailed analysis of the boundary layers for mixed hyperbolic‐parabolic systems in one space dimension and small amplitude regimes. As an application of our results, we describe the solution of the so‐called boundary Riemann problem recovered as the zero viscosity limit of the physical viscous approximation. In particular, we tackle the so‐called doubly characteristic case, which is considerably

This paper investigates the stability of traveling wave solutions to the free boundary Euler equations with a submerged point vortex. We prove that sufficiently small‐amplitude waves with small enough vortex strength are conditionally orbitally stable. In the process of obtaining this result, we develop a quite general stability/instability theory for bound state solutions of a large class of infinite‐dimensional

We exploit the so‐called atomic condition, recently defined by De Philippis, De Rosa, and Ghiraldin and proved to be necessary and sufficient for the validity of the anisotropic counterpart of the Allard rectifiability theorem. In particular, we address an open question of this seminal work, showing that the atomic condition implies the strict Almgren geometric ellipticity condition. © 2020 Wiley Periodicals

We investigate in this work a recently proposed diagrammatic quantum Monte Carlo method—the inchworm Monte Carlo method—for open quantum systems. We establish its validity rigorously based on resummation of Dyson series. Moreover, we introduce an integro‐differential equation formulation for open quantum systems, which illuminates the mathematical structure of the inchworm algorithm. This new formulation

We investigate the homogeneous Dirichlet problem for the fast diffusion equation ut = Δum, posed in a smooth bounded domain Ω ⊂ ℝN, in the exponent range ms = (N − 2)+/(N + 2) < m < 1. It is known that bounded positive solutions extinguish in a finite time T > 0, and also that they approach a separate variable solution u(t, x) ∼ (T − t)1/(1 − m)S(x) as t → T−, where S belongs to the set of solutions

A spectral approach to building the exterior calculus in manifold learning problems is developed. The spectral approach is shown to converge to the true exterior calculus in the limit of large data. Simultaneously, the spectral approach decouples the memory requirements from the amount of data points and ambient space dimension. To achieve this, the exterior calculus is reformulated entirely in terms

We consider the Ising model at its critical temperature with external magnetic field ha15/8 on the square lattice with lattice spacing a. We show that the truncated two‐point function in this model decays exponentially with a rate independent of a as a ↓ 0. As a consequence, we show exponential decay in the near‐critical scaling limit Euclidean magnetization field. For the lattice model with a = 1

We study inverse problems for the Einstein equations with source fields in a general form. Under a microlocal linearization stability condition, we show that by generating small gravitational perturbations and measuring the responses near a freely falling observer, one can uniquely determine the background Lorentzian metric up to isometries in a region where the gravitational perturbations can travel

We study the solvability of second boundary value problems of fourth‐order equations of Abreu type arising from approximation of convex functionals whose Lagrangians depend on the gradient variable, subject to a convexity constraint. These functionals arise in different scientific disciplines such as Newton's problem of minimal resistance in physics and the monopolist's problem in economics. The right‐hand

We consider nonuniformly elliptic variational problems and give optimal conditions guaranteeing the local Lipschitz regularity of solutions in terms of the regularity of the given data. The analysis catches the main model cases in the literature. Integrals with fast, exponential‐type growth conditions as well as integrals with unbalanced polynomial growth conditions are covered. Our criteria involve

We consider a subclass of those quasilinear wave equations in 3 + 1 space‐time dimensions that satisfy the “weak null condition” as defined by Lindblad and Rodnianski , and study the large‐time behavior of solutions to the Cauchy problem. The prototype for the class of equations considered is . Global solutions for such equations have been constructed by Lindblad and Alinhac. Our main results are the

A classical problem in fluid mechanics is the motion of an axisymmetric vortex sheet evolving under the action of surface tension, surrounded by an inviscid fluid. Lagrangian descriptions of these dynamics are well‐known, involving complex nonlocal expressions for the radial and longitudinal velocities in terms of elliptic integrals. Here we use these prior results to arrive at a remarkably compact

We study the following control problem. A fish with bounded aquatic locomotion speed swims in fast waters. Can this fish, under reasonable assumptions, get to a desired destination? It can, even if the flow is time dependent. Moreover, given a prescribed sufficiently large time t, it can be there at exactly the time t. The major difference from our previous work is the time dependence of the flow.