arXiv research

The paper improves energy decay estimates for Dir-stationary Q-valued functions and applies them to Liouville-type theorems and continuity.

New inequality shows energy growth and decay in geometric problems.

Removes singularities for Yang-Mills-Higgs fields in higher dimensions.

Paper uses 3-circle theorem to study Willmore surfaces and prove decay estimates.

Many practical machine learning tasks employ very deep convolutional neural networks. Such large depths pose formidable computational challenges in training and operating the network. It is therefore important to understand how fast the energy contained in the propagated signals (a.k.a. feature maps) decays across laye…

Second part of series studying charged scalar fields on Reissner--Nordström spacetimes.

Study on finite energy SU(2) monopoles on AC 3-manifolds, proving integrality of charge and curvature decay.

New energy functional bounds Ricci flows on ancient spaces.

The paper studies decay near singularities of 3d Yang-Mills-Higgs fields.

In recent work, we have proven uniform decay bounds for solutions of the wave equation $\Box_gφ=0$ on a Schwarzschild exterior, in particular, the uniform pointwise estimate $|φ|\le Cv_+^{-1}$, which holds throughout the domain of outer communications, where $v$ is an advanced Eddington-Finkelstein coordinate, $v_+=\ma…

The study examines asymptotic properties of G2-monopoles on nonparabolic G2-manifolds.

We establish that finite-time singularities do not occur in four-dimensional Yang-Mills flow, confirming the conjecture of Schlatter, Struwe, and Tahvildar-Zadeh. The proof relies on a weighted energy identity and sharp decay estimates in the neck region.

Study proposes a nonlocal approximation of the Willmore functional using fractional Allen-Cahn energies.

Study axisymmetric waves on extremal Kerr spacetime using physical-space estimates.

The paper establishes scattering theory for wave equations on Schwarzschild spacetime.

This paper contains the first two parts (I-II) of a three-part series concerning the scalar wave equation \Box_gψ = 0 on a fixed Kerr background. We here restrict to two cases: (II1) |a| \ll M, general ψ or (II2) |a| < M, ψ axisymmetric. In either case, we prove a version of 'integrated local energy decay', specificall…

Study of charged scalar fields on Reissner-Nordström spacetimes via energy estimates.

We study the Cauchy problem for the wave equation on extreme Kerr backgrounds under axisymmetry. Specifically, we consider regular axisymmetric initial data prescribed on a Cauchy hypersurface S which connects the future event horizon with spacelike or null infinity, and we solve the linear wave equation on the domain …

We derive a selection of energy estimates for a generalisation of a critical equation on the unit disc in $\mathbb{R}^2$ introduced by Rivière. Applications include sharp regularity results and compactness theorems which generalise a large amount of previous geometric PDE theory, including some of the theory of harmoni…

This paper contains the second part of a two-part series on the stability and instability of extreme Reissner-Nordstrom spacetimes for linear scalar perturbations. We continue our study of solutions to the linear wave equation on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial da…

We give conditions on a general stress-energy tensor T_{αβ} in a spherically symmetric black hole spacetime which are sufficient to guarantee that the black hole will contain a (spherically symmetric) marginally trapped tube which is eventually achronal, connected, and asymptotic to the event horizon. Price law decay p…

New proof removes decay assumptions for spacetime positive mass theorem.

Study proves global existence and decay for complex wave equations.

This is the second in a series of three papers in which we initiate the study of very rough solutions to the initial value problem for the Einstein vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and Sob…

Belief propagation is a fundamental message-passing algorithm for probabilistic reasoning and inference in graphical models. While it is known to be exact on trees, in most applications belief propagation is run on graphs with cycles. Understanding the behavior of "loopy" belief propagation has been a major challenge f…

New proof shows perturbed non-compact Einstein spaces attract to unique global solution.

The fundamental properties of $J$-holomorphic maps depend on two inequalities: The gradient inequality gives a pointwise bound on the differential of a $J$-holomorphic map in terms of its energy. The cylinder inequality stipulates and quantifies the exponential decay of energy along cylinders of small total energy. We …

In this paper, we prove that the Schrödinger map flows from $\Bbb R^d$ with $d\ge 3$ to compact Kähler manifolds with small initial data in critical Sobolev spaces are global. This is a companion work of our previous paper [23] where the energy critical case $d=2$ was solved. In the first part of this paper, for heat f…

Let $X=\mathbb{C}\timesΣ$ be the product of the complex plane and a compact Riemann surface. We establish a classification theorem of solutions to the Seiberg-Witten equation on $X$ with finite analytic energy. The spin bundle $S^+\to X$ splits as $L^+\oplus L^-$. When $2-2g\leq c_1(S^+)[Σ]<0$, the moduli space is in b…

Study on massless Vlasov equation on Reissner-Nordström spacetimes, showing decay rates and non-decay phenomena.

Study on helix curves and their Möbius energy asymptotics.

We consider solutions to the linear wave equation $\Box_gφ=0$ on a non-extremal maximally extended Schwarzschild-de Sitter spacetime arising from arbitrary smooth initial data prescribed on an arbitrary Cauchy hypersurface. (In particular, no symmetry is assumed on initial data, and the support of the solutions may con…

We study the convergence of the Kähler-Ricci flow on a Fano manifold under some stability conditions. More precisely we assume that the first eingenvalue of the $\bar\partial$-operator acting on vector fields is uniformly bounded along the flow, and in addition the Mabuchi energy decays at most logarithmically. We then…

Study examines wave equation decay and Strichartz estimates on conic manifolds.

The paper extends decay estimates to graphs with positive spectrum.

The free energy is a key quantity of interest in Ising models, but unfortunately, computing it in general is computationally intractable. Two popular (variational) approximation schemes for estimating the free energy of general Ising models (in particular, even in regimes where correlation decay does not hold) are: (i)…

In the early 1980's Almgren developed a theory of Dirichlet energy minimizing multi-valued functions, proving that the Hausdorff dimension of the singular set (including branch points) of such a function is at most $(n-2),$ where $n$ is the dimension of its domain. Almgren used this result in an essential way to show t…

A common problem in a high energy physics experiment is extracting a signal from a much larger background. Posed as a classification task, there is said to be an imbalance in the number of samples belonging to the signal class versus the number of samples from the background class. In this work we provide a brief overv…

This is the second in a series of papers in which we take a systematic study of gauge field theories such as the Maxwell equations and the Yang-Mills equations, on curved space-times. In this paper, we study the Maxwell equations in the domain of outer-communication of the Schwarzschild black hole. We show that if we a…

In this global study of solutions to the linear wave equation on Schwarzschild de Sitter spacetimes we attend to the cosmological region of spacetime which is bounded in the past by cosmological horizons and to the future by a spacelike hypersurface at infinity. We prove an energy estimate capturing the expansion of th…

Global existence and decay for quasilinear wave equations on various spacetimes, including Kerr black holes.

The stability of physical systems depends on the existence of a state of least energy. In gravity, this is guaranteed by the positive energy theorem. For topological reasons this fails for nonsupersymmetric Kaluza-Klein compactifications, which can decay to arbitrarily negative energy. For related reasons, this also fa…

Extends global stability of Minkowski spacetime to minimal decay assumptions.

We investigate Yang--Mills instanton theory over four dimensional asymptotically locally flat (ALF) geometries, including gravitational instantons of this type, by exploiting the existence of a natural smooth compactification of these spaces introduced by Hausel--Hunsicker--Mazzeo. First referring to the codimension 2 …

Study optimizes decay estimates for minimizing currents in submanifolds.

This is the first in a series Of papers in which we initiate the study Of very rough solutions to the initial value problem for the Einstein Vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques Of energy estimates and Sobolev in…

Study finds solutions to inequality decay to zero on warped cylinders.

New decay estimates for scalar curvature of steady gradient Ricci solitons.