arXiv research

New inequality shows energy growth and decay in geometric problems.

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

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

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…

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…

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

New proof removes decay assumptions for spacetime positive mass theorem.

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…

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

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

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 …

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

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 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…

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

New energy functional bounds Ricci flows on ancient spaces.

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.

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…

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…

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 …

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…

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

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…

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

New magnetic memory effects found in gravitational waves and memory.

Survey on stability of Minkowski spacetime in relativity.

Future stability of FLRW solutions in expanding 3D space is shown for compact perturbations.

We propose the time-dependent generalization of an `ordinary' autonomous human biomechanics, in which total mechanical + biochemical energy is not conserved. We introduce a general framework for time-dependent biomechanics in terms of jet manifolds derived from the extended musculo-skeletal configuration manifold. The …

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 insights into Bartnik mass from improvability of dominant energy scalar.

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…

Physics-informed learning framework for pH systems and EB-PBC control.

We extend the Jang equation proof of the positive energy theorem due to R. Schoen and S.-T. Yau from dimension $n=3$ to dimensions $3 \leq n <8$. This requires us to address several technical difficulties that are not present when $n=3$. The regularity and decay assumptions for the initial data sets to which our argume…

Study on helix curves and their Möbius energy asymptotics.

Growth of spinors in 4D and 3D generalized Seiberg-Witten equations.

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

New energy definition for expanding de Sitter spacetime with umbilic boundaries.

New method for LVEBMs using saddle-point optimization and Langevin updates.

Paper proves rigidity theorems for AE Q-singular spaces.

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

We develop a definitive physical-space scattering theory for the scalar wave equation on Kerr exterior backgrounds in the general subextremal case |a|<M. In particular, we prove results corresponding to "existence and uniqueness of scattering states" and "asymptotic completeness" and we show moreover that the resulting…

Study quadratic one-forms on logarithmic Higgs bundles on pointed curves.

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…

A new flow method reduces Lorentz contraction to a simple algebraic decay.

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

Study proves global existence and decay for complex wave equations.