arXiv research

The paper proves conditions under which critical point metrics are Einstein.

Ricci solitons as critical points of quadratic curvature functionals

New metrics found in hyperbolic manifolds as volume-minimizers.

Study stability of non-Kähler Calabi-Yau metrics using critical points of generalized Einstein Hilbert action.

Paper proves conjecture about Einstein metrics on manifolds with positive isotropic curvature.

New findings on Chern flat metrics and their criticality.

Rigidity theorem for special metrics on 4-manifolds.

In this paper, we investigate critical points of the Laplacian's eigenvalues considered as functionals on the space of Riemmannian metrics or a conformal class of metrics on a compact manifold. We obtain necessary and sufficient conditions for a metric to be a critical point of such a functional. We derive specific con…

The article studies critical points of a new energy functional in higher dimensions.

The paper proves new rigidity results for critical metrics of quadratic curvature functionals.

We study rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals involving the scalar curvature, the Ricci curvature and the Riemannian curvature tensor, characterized by some pointwise inequalities involving the Weyl curvature and the traceless Ricci curvat…

On the space of positive 3-forms on a seven-manifold, we study a natural functional whose critical points induce metrics with holonomy contained in $G_2$. We prove short-time existence and uniqueness for its negative gradient flow. Furthermore, we show that the flow exists for all times and converges modulo diffeomorph…

New framework for studying eigenvalue functionals of metrics.

Study on metrics maximizing eigenvalues of Paneitz operator on 4-manifolds.

We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In…

The study counts critical points of Steklov eigenfunctions on manifolds.

The study examines metrics with unit volume or area on manifolds with boundaries, finding critical points and solving curvature problems.

New approach finds Kähler metrics on compact complex manifolds.

Let C be a hyperelliptic Riemann surface. We show that the hyperelliptic Weierstrass points of C are non-degenerated critical points of Morse index +2 of the curvature function K of the Theta metric on C (called also Bergman metric). When the genus of C is two, we compute all critical points of K and we show in this ca…

The paper introduces a volume invariant for Hermitian-symplectic metrics and proves its critical points are Kähler.

In this paper, we consider some rigidity results for the Einstein metrics as the critical points of some known quadratic curvature functionals on complete manifolds, characterized by some point-wise inequalities. Moreover, we also provide rigidity results by the integral inequalities involving the Weyl curvature, the t…

In this paper, we prove some rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals on closed manifolds, characterized by some point-wise inequalities. Moreover, we also provide a few rigidity results that involve the Weyl curvature, the trace-less Ricci cu…

In this paper we prove rigidity results on critical metrics for quadratic curvature functionals, involving the Ricci and the scalar curvature, on the space of Riemannian metrics with unit volume. It is well-known that Einstein metrics are always critical points. The purpose of this article is to show that, under some c…

Adapted metrics found on complex manifolds.

Let $\mathcal{C}$ be the space of smooth metrics $g$ on a given compact manifold $M^{n}$ ($n\geq3$) with constant scalar curvature and unitary volume. The goal of this paper is to study the critical point of the total scalar curvature functional restricted to the space $\mathcal{C}$ (we shall refer to this critical poi…

The exponential map fails to be injective near critical points in sub-Riemannian geometry.

Solves Besse conjecture on 3D manifolds, proving metric rigidity.

On a given closed connected manifold of dimension two, or greater, we consider the squared $L^2$-norm of the scalar curvature functional over the space of constant volume Riemannian metrics. We prove that its critical points have constant scalar curvature, and use this to show that a metric is a solution of the critica…

We investigate rigidity and stability properties of critical points of quadratic curvature functionals on the space of Riemannian metrics. We show it is possible to "gauge" the Euler-Lagrange equations, in a self-adjoint fashion, to become elliptic. Fredholm theory may then be used to describe local properties of the m…

We study the space of smooth Riemannian structures on compact three-manifolds with boundary that satisfies a critical point equation associated with a boundary value problem, for simplicity, Miao-Tam critical metrics. We provide an estimate to the area of the boundary of Miao-Tam critical metrics on compact three-manif…

Non-degeneracy of critical points proven for manifold's squared norm of second fundamental form.

On a compact n-dimensional manifold, it has been conjectured that a critical point metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, will be Einstein. This conjecture was proposed in 1984 by Besse, but has yet to be proved. In this paper, we prove th…

Paper derives second variation formula for eigenvalue functionals on surfaces.

The paper classifies contact 3-manifolds with critical metrics and connects entropy to optimization.

Study finds critical points in perimeter functional for fixed volume sets.

In [LS], it is shown shown that the first eigenvalue of the Laplacian restricted to the space of invariant functions on a toric Kähler manifold (i.e. $λ_1^\mathbb{T}$, the invariant first eigenvalue) is an unbounded function of the toric Kähler metric. In this note we show that, seen as a function on the space of toric…

Extends Gauduchon's result to higher dimensions, showing balanced metrics.

Study moment maps coupled with convex functions to find critical points.

We prove that a Kahler metric in the anticanonical class which is a critical point of the functional E_k and has nonnegative Ricci curvature, is necessarily Kahler-Einstein. This partially answers a question of X.X.Chen.

We solve the classifying problem raised by Fischer and Marsden for Bach flat static spaces. We also prove the conjecture about critical point equations proposed by Besse for Bach flat manifolds. Particularly in dimension 3, we derive an integral identity that allows us to obtain conformal flatness from the vanish of th…

The purpose of this paper is to investigate the critical points of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume, for simplicity CPE metrics. It was conjectured in $1980$'s that every CPE metric must be Einstein. We prove that a $4$-dimensional CPE…

The conformal properties of complex Finsler metrics are studied. We give a characterization of a compact complex Finsler manifold to be globally conformal Kähler. The critical points of the total holomorphic curvature and total Ricci curvature in the volume preserved conformal classes are studied. The stability of crit…

Study the stability of Einstein metrics on homogeneous spaces.

We define a quantisation of the J-flow over a projective complex manifold. As corollaries, we obtain new proofs of uniqueness of critical points of the J-flow and that these critical points achieve the absolute minimum of an associated energy functional. We show that the existence of a critical point of the J-flow impl…

Abstract: Characterizes special Kähler manifolds with specific properties.

We show stability of pairs of Ricci flat metrics and parallel spinor fields with respect to the spinor flow, i.e. we show that the spinor flow with initial conditions near such pairs converges to a critical point with exponential speed. Moreover, we show stability of certain volume constrained critical points of the sp…

Several rigidity results are proved for critical points of natural Riemannian functionals on the space of metrics on 3-manifolds. Two of these results are as follows. Let (N, g) be a complete Riemannian 3-manifold, satisfying one of the following variational conditions: (i) (N, g) has non-negative scalar curvature and …

The paper studies metrics that match prescribed geodesics and introduces a variational problem.