arXiv research

The paper studies $m$-quasi Einstein manifolds with convex potential and finds constant scalar curvature.

Proves global existence and uniqueness of solutions for Einstein-scalar-field equations.

New method trains normalizing flows using entropy-regularized transport.

Compactness theorems for $G_2$-solitons established with scalar curvature and potential function constraints.

We derive a sharp lower bound for the scalar curvature of non-flat and non-compact expanding gradient Ricci soliton provided that the scalar curvature is non-negative and the potential function is proper. We also give an upper bound for the scalar curvature of noncompact expander when the Ricci curvature is nonpositive…

Characterizes gradient Yamabe solitons with specific conditions.

The study proves unique static manifolds with positive scalar curvature and boundary.

The paper proves inequalities and growth rates for Schouten solitons.

We explain how to derive largeness constraints in scalar curvature geometry using some basic splitting results and the potential theory on singular area minimizing hypersurfaces. This includes a variety of results like the non-existence of positive scalar curvature metrics on enlargeable manifolds or simplified proofs …

The paper characterizes solitons and estimates scalar curvature.

The study characterizes quasi Yamabe solitons with potential vector fields.

If the potential vector field of an $η$-Ricci soliton is of gradient type, using Bochner formula, we derive from the soliton equation a Laplacian equation satisfied by the potential function $f$. In a particular case of irrotational potential vector field we prove that the soliton is completely determined by $f$. We gi…

This note relaxes conditions for Kähler metrics with bounded entropy and scalar curvature.

Paper refines Chen-Cheng's estimates for Kähler metrics.

Study on rigidity and characterization of generalized quasi-Einstein manifolds.

In this paper we study potential function of gradient steady Ricci solitons. We prove that infimum of potential function decays linearly; in particular, potential function of rectifiable gradient steady Ricci solitons decays linearly. As a consequence, we show that a gradient steady Ricci soliton with bounded potential…

A holomorphy potential is a complex valued function whose complex gradient, with respect to some Kähler metric, is a holomorphic vector field. Given $k$ holomorphic vector fields on a compact complex manifold, form, for a given Kähler metric, a product of the following type: a function of the scalar curvature multiplie…

We study the influence of an additional scalar potential on various geometric and analytic properties of Dirac-harmonic maps. We will create a mathematical wish list of the possible benefits from inducing the potential term and point out that the latter cannot be achieved in general. Finally, we focus on several potent…

Study shows inflation in 3+1D cosmologies with bounded scalar potential and specific symmetry.

In this paper, we study dynamics of geodesic flows over closed surfaces of genus greater than or equal to 2 without focal points. Especially, we prove that there is a large class of potentials having unique equilibrium states, including scalar multiples of the geometric potential, provided the scalar is less than 1. Mo…

5D shrinking Ricci solitons with constant scalar curvature are rigid.

We consider, in the Euclidean setting, a conformal Yamabe-type equation related to a potential generalization of the classical constant scalar curvature problem and which naturally arises in the study of Ricci solitons structures. We prove existence and nonexistence results, focusing on the radial case, under some gene…

New proof of Penrose inequality using potential theory.

Starting with a model conical Kähler metric, we prove a uniform scalar curvature bound for solutions to the conical Kähler-Ricci flow assuming a semi-ampleness type condition on the twisted canonical bundle. In the proof, we also establish uniform estimates for the potentials and their time derivatives.

In dimension $4$, we show that a nontrivial flat cone cannot be approximated by smooth Ricci shrinkers with bounded scalar curvature and Harnack inequality, under the pointed-Gromov-Hausdorff topology. As applications, we obtain uniform positive lower bounds of scalar curvature and potential functions on Ricci shrinker…

We summarize the global geometric formulation of Einstein-Scalar-Maxwell theories twisted by flat symplectic vector bundle which encodes the duality structure of the theory. We describe the scalar-electromagnetic symmetry group of such models, which consists of flat unbased symplectic automorphisms of the flat symplect…

Study on Einstein solitons with bounds and asymptotic behavior.

In this paper, we derive apriori estimates for constant scalar curvature Kähler metrics on a compact Kähler manifold. We show that higher order derivatives can be estimated in terms of a $C^0$ bound for the Kähler potential. We also discuss some local versions of these estimates which can be of independent interest.

Study of para-Ricci-like solitons on specific Riemannian manifolds.

In this paper we show that every area minimizing cone C^{n-1} in R^n can be approximated by entirely smooth area minimizing hypersurfaces. This extensively uses hyperbolic unfoldings of such hypersurfaces and the resulting potential theory for the Jacobi field operator. Applications include the splitting theorem in sca…

The paper derives estimates for linear potentials and applies them to improve Hausdorff dimensions of singular sets in conformal geometry.

We derive lower bounds on the scalar curvature of complete non-compact gradient Yamabe solitons under some integral curvature conditions. Based on this, we prove that the corresponding potential functions have at most quadratic growth in distance. We also obtain a finite topological type property on complete shrinking …

Paper defines Bartnik mass for hyperbolic extensions and proves staticity.

Quantum kernel machines need to use more complex kernels to fully exploit their potential.

Kähler soliton surfaces are typically toric under generic conditions.

Paper proves rigidity for Ricci solitons with specific conditions.

This paper deals with the conformal deformation of the standard metric in a domain on the sphere to a complete metric with the constant scalar curvature. The problem of description of domains allowing such deformation originates in the works of Loewner and Nirenberg, and Schoen and Yau concerned with the locally confor…

The study preserves lower bounds of total scalar curvature under specific metric convergence.

In this paper, we study gradient Ricci expanding solitons $(X,g)$ satisfying $Rc=cg+D^2f,$ where $Rc$ is the Ricci curvature, $c<0$ is a constant, and $D^2f$ is the Hessian of the potential function $f$ on $X$. We show that for a gradient expanding soliton $(X,g)$ with non-negative Ricci curvature, the scalar curva…

We consider spacetimes solving the Einstein non-linear scalar field equations with T2-symmetry and show that they admit an areal time foliation in the expanding direction. In particular, we prove global existence and uniqueness of solutions to the corresponding system of evolution equations for all future times. The on…

Solutions to scalar curvature equations have the property that all possible blow-up points are isolated, at least in low dimensions. This property is commonly used as the first step in the proofs of compactness. We show that this result becomes false for some arbitrarily small, smooth perturbations of the potential.

The study shows ends of shrinking gradient $ρ$-Einstein solitons are non-parabolic.

New bounds for Dirac eigenvalue involving boundary capacity.

In this paper we reformulate N=2 supergravity backgrounds arising in type II string theory in terms of quantities transforming under the U-duality group E7(7). In particular we combine the Ramond--Ramond scalar degrees of freedom together with the O(6,6) pure spinors which govern the Neveu-Schwarz sector by considering…

Let $N$ be a closed enlargeable manifold in the sense of Gromov-Lawson and $M$ a closed spin manifold of equal dimension, a famous theorem of Gromov-Lawson states that the connected sum $M\# N$ admits no metric of positive scalar curvature. We present a potential generalization of this result to the case where $M$ is n…

In this paper, we consider the $V$-soliton equation which is a degenerate fully nonlinear equation introduced by La Nave and Tian in their work on Kähler-Ricci flow on symplectic quotients. One can apply the interpretation to study finite time singularities of the Kähler-Ricci flow. As in the case of Kähler-Einstein me…

We study the eigenvalues of the magnetic Schroedinger operator associated with a magnetic potential A and a scalar potential q, on a compact Riemannian manifold M, with Neumann boundary conditions if the boundary is not empty. We obtain several bounds for the spectrum. Besides the dimension and the volume of the manifo…

In a recent paper Donaldson explains how to use an older construction of Joyce to obtain four dimensional local models for scalar-flat Kahler metrics with a 2-torus symmetry. Using this idea, he recovers and generalizes the Taub-NUT metric by including it in a new family of complete scalar-flat toric Kahler metrics. In…