arXiv research

We establish the positive energy theorem for weak asymptotically anti-de Sitter initial data sets with distributional curvature under the weak dominant energy condition.

We prove global existence for solutions arising from small initial data for a large class of quasilinear wave equations satisfying the `weak null condition' of Lindblad and Rodnianski, significantly enlarging upon the class of equations for which global existence is known. In addition to the usual weak null condition, …

We consider localized deformation for initial data sets of the Einstein field equations with the dominant energy condition. Deformation results with the weak inequality need to be handled delicately. We introduce a modified constraint operator to absorb the first order change of the metric in the dominant energy condit…

The paper finds exact solutions to a complex Einstein-Dirac-Maxwell system on 4D Sasakian spacetimes.

Paper proves uniqueness of weak solutions for Plateau flow.

Study of Dirac-Witten operator on Lorentzian manifolds under dominant energy condition.

Defines weak geodesics on specific subsets of manifolds.

The paper proves optimal estimates and inequalities for spectral functions on certain manifolds.

In this paper, we prove that a sequence of weak almost Kähler-Ricci solitons under further suitable conditions converge to a Kähler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology. As a corollary, we show that on a Fano manifold with the modified K-energy bounded belo…

We show that the Hawking--Penrose singularity theorem, and the generalisation of this theorem due to Galloway and Senovilla, continue to hold for Lorentzian metrics that are of $C^{1, 1}$-regularity. We formulate appropriate weak versions of the strong energy condition and genericity condition for $C^{1,1}$-metrics, an…

We prove existence and regularity of minimisers for the Canham-Helfrich energy in the class of weak (possibly branched and bubbled) immersions of the $2$-sphere. This solves (the spherical case) of the minimisation problem proposed by Helfrich in 1973, modelling lipid bilayer membranes. On the way to prove the main res…

We will give a weak energy identity for Sacks-Uhlenbeck approximation of harmonic maps and calculate the length of the necks.

A new parametric method studies Willmore flows and energy quantization.

Quantizes Willmore energy in Riemannian manifolds with bounded energy and area.

We develop some pluripotential theoretic techniques for the transversally holomorphic foliation of a Sasakian manifold. We prove the convexity of the K-energy along weak geodesics for Sasakian manifolds. This implies that the K-energy is bounded below if a constant scalar curvature structure exists with those metrics m…

Study of Kähler-Ricci flow on toric Fano varieties with preserved symplectic condition.

In this paper, we discuss a Donaldson's version of the modified $K$-energy associated to the Calabi's extremal metrics on toric manifolds and prove the existence of the weak solution for extremal metrics in the sense of convex functions which minimizes the modified $K$-energy.

We are analysing the convexity and continuity properties of the Mabuchi functional along weak geodesics. The key technical point in our paper is the global approximation of weak geodesics obtained via a well-chosen family of Monge-Ampère equations.

We prove that the critical points of various energies such as the area, the Willmore energy, the frame energy for tori...etc among possibly branched immersions constrained to evolve within a smooth sub-manifold of the Teichmüller space satisfy the corresponding constrained Euler Lagrange equation. We deduce that critic…

Defines weak normals for irregular curves in high-dimensional spaces.

Study well-poses Dirac operator problem with APS boundary conditions.

Analyzes perturbed contact instantons with Legendrian boundary conditions using geometric analysis.

Unified approach to various energy conditions in spacetime geometry.

New elastic energy for irregular curves defined through polygonal approximations.

The integral of the energy density function $\mathfrak m$ of a closed Robertson-Walker (RW) spacetime with source a perfect fluid and cosmological constant $Λ$ gives rise to an action functional on the space of scale functions of RW spacetime metrics. This paper studies closed RW spacetimes which are critical for this …

Let $(X,ω)$ be a compact connected Kähler manifold and denote by $(\mathcal E^p,d_p)$ the metric completion of the space of Kähler potentials $\mathcal H_ω$ with respect to the $L^p$-type path length metric $d_p$. First, we show that the natural analytic extension of the (twisted) Mabuchi K-energy to $\mathcal E^p$ is …

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…

The paper proves properties of curves in Riemannian manifolds.

As sensor networks for health monitoring become more prevalent, so will the need to control their usage and consumption of energy. This paper presents a method which leverages the algorithm's performance and energy consumption. By utilising Reinforcement Learning (RL) techniques, we provide an adaptive framework, which…

We study the minimization problem for the Yang-Mills energy under fixed boundary connection in supercritical dimension $n\geq 5$. We define the natural function space A_{G} in which to formulate this problem in analogy to the space of integral currents used for the classical Plateau problem. The space A_{G} can be also…

We establish the convexity of Mabuchi's K-energy functional along weak geodesics in the space of Kahler potentials on a compact Kahler manifold thus confirming a conjecture of Chen and give some applications in Kahler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally …

We consider the energy and bienergy functionals as variational problems on the set of Riemannian metrics and present a study of the biharmonic stress-energy tensor. This approach is then applied to characterise weak conformality of the Gauss map of a submanifold. Finally, working at the level of functionals, we recover…

Study proves global existence and decay for complex wave equations.

We show that K-energy minimizing movements agree with smooth solutions to Calabi flow as long as the latter exist. As corollaries we conclude that in a general Kahler class long time solutions of Calabi flow minimize both K-energy and Calabi energy. Lastly, by applying convergence results from the theory of minimizing …

The (twice-contracted) second Bianchi identity is a differential curvature identity that holds on any smooth manifold with a metric. In the case when such a metric is Lorentzian and solves Einstein's equations with an (in this case inevitably smooth) energy-momentum-stress tensor of a "matter field" as the source of sp…

In this paper we study constant scalar curvature equation (CSCK), a nonlinear fourth order elliptic equation, and its weak solutions on Kähler manifolds. We first define a notion of weak solution of CSCK for an $L^\infty$ Kähler metric. The main result is to show that such a weak solution (with uniform $L^\infty$ bound…

Explain convexity of K-energy leading to unique metrics.

The study characterizes spacetime and modified gravity models using projective curvature tensor.

Study calculates the elastic energy of curves on a sphere.

Study on fourth order Lamm-Riviere system for biharmonic mappings in 4D.

Critical points of scale-invariant curvature energies in 4D are analytic.

In this paper, we consider a parabolic system from a bounded domain in a Euclidean space or a closed Riemannian manifold into a unit sphere in a compact Lie algebra $\mathfrak{g}$, which can be viewed as the extension of Landau-Lifshtiz (LL) equation and was proposed by V. Arnold. We follow the ideas taken from the wor…

The geodesic equation for the right invariant $L^2$-metric (which is a weak Riemannian metric) on each Virasoro-Bott group is equivalent to the KdV-equation. We prove that the corresponding energy functional, when restricted to paths with fixed endpoints, has no local minima. In particular solutions of KdV don't define…

Introduces weak $(p,k)$-Dirac structures in geometric settings.

Study on dynamic curves with elastic energy and spontaneous curvature.

A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed. The equation is reformulated as a conservation law and solved by a suitable Ginzburg-Landau type approximation.

In this paper, we continue to study the Calabi flow on complex tori. We develop a new method to obtain an explicit bound of the curvature of the Calabi flow. As an application, we show that when $n=2$, the Calabi flow starting from a weak Kähler metric will become smooth immediately. It implies that in our settings, th…

This work optimizes DNN inference for energy-harvesting devices by compressing and selectively executing neural network exits.