arXiv research

Generalizes Aubin's result for Yamabe-type problem on smooth metric measure spaces.

We prove Aubin's "Hypothese fondamentale" concerning the existence of Moser-Trudinger type inequalities on any integral compact Kähler manifold X. In the case of the anti-canonical class on a Fano manifold the constants in the inequalities are shown to only depend on the dimension of X (but there are counterexamples to…

Study Dirac-Einstein equations on manifolds with boundary, focusing on constant volume and chiral conditions.

Study on Palais-Smale sequences for conformal Dirac-Einstein problem, proving existence of solutions.

Study of Dirac equation with non-local nonlinearity on spheres.

New proof of Aubin-Yau theorem for complex non-Kähler manifolds.

The paper constructs metrics on compact manifolds using Aubin's deformations.

The paper generalizes inequalities on almost Kähler manifolds.

In this note we give a detailed proof of a theorem of Aubin.

Generalizes Escobar-Riemann mapping problem for smooth metric measure spaces.

Solves complex Monge-Ampère equation for Kähler-Ricci solitons.

In the previous papers \cite{L1, L2} the author constructed Mabuchi and Aubin-Yau functionals over any complex surfaces and three-folds, respectively. Using the method in \cite{L2}, we construct those functionals over any complex manifolds of the complex dimension bigger than or equal to 2.

In this note we construct Mabuchi $\mathcal{L}^{\rm M}_ω$ functional and Aubin-Yau functionals $\mathcal{I}^{\rm AY}_ω, \mathcal{J}^{\rm AY}_ω$ on any compact complex surfaces, and establish a number of properties. Our construction coincides with the original one in the Kähler case.

In their study of the Yamabe problem in the presence of isometry group, Hebey and Vaugon announced a conjecture. This conjecture generalizes Aubin's conjecture, which has already been proven and is sufficient to solve the Yamabe problem. In this paper, we generalize Aubin's theorem and we prove the Hebey--Vaugon conjec…

In this paper we construct Mabuchi $\mathcal{L}^{\rm M}_ω$ functional and Aubin-Yau functionals $\mathcal{I}^{\rm AY}_ω, \mathcal{J}^{\rm AY}_ω$ on any compact complex three-folds. The method presented here will be used in the forthcoming paper \cite{L1} on the construction of those functionals on any compact complex m…

We derive an explicit formula for the asymptotic slope of the Aubin-Yau functional along a Bergman geodesic on a surface of complex dimension 2, extending the work of Phong-Sturm on Riemann surfaces. This is equivalent to an explicit calculation of the Donaldson-Futaki invariant of a test configuration. The slope is gi…

New Kähler metrics found with cone singularities on complex manifolds.

Proves existence of Yamabe metrics on conical manifolds with conical points and links.

Solves complex equation for specific geometric solitons.

Improved Moser-Trudinger-Onofri inequality with constraints on sphere.

Smooth solutions and classification of Dirac-Einstein equations on 3-manifolds.

The paper examines stability of the Sobolev inequality in metric spaces with curvature dimension conditions.

We study complex Monge-Ampere equations on Hermitian manifolds, extending classical existence results of Yau and Aubin in the Kahler case, and those of Caffarelli, Kohn, Nirenberg and Spruck for the Dirichlet problem in $C^n$. As an application we generalize existing results on the Donaldson conjecture on geodesics in …

New flow on compact manifolds solves Yamabe equation.

Simplified proof and new $C^0$ estimate for Kähler-Einstein metrics.

Let $(M,g,\si)$ be a compact Riemannian spin manifold of dimension $\geq 2$. For any metric $\tilde g$ conformal to $g$, we denote by $\tildeλ$ the first positive eigenvalue of the Dirac operator on $(M,\tilde g,\si)$. We show that $$\inf_{\tilde{g} \in [g]} \tildeλ\Vol(M,\tilde g)^{1/n} \leq (n/2) \Vol(S^n)^{1/n}.$$ T…

The paper classifies Cartan-Hadamard manifolds supporting optimal Sobolev inequalities.

We study the blowup behavior at infinity of the normalized Kahler-Ricci flow on a Fano manifold which does not admit Kahler-Einstein metrics. We prove an estimate for the Kahler potential away from a multiplier ideal subscheme, which implies that the volume forms along the flow converge to zero locally uniformly away f…

Study on pseudo-Einstein 3-manifolds for a specific inequality, introducing Robin mass.

In the first part of this thesis, we study the Yamabe problem with singularities, that we can announce as follow: Given a compact Riemannian manifold $(M,g)$, find a constant scalar curvature metric, conformal to $g$, when $g$ has not necessarily the usual regularity (it can be $C^1$). To solve this problem, we start t…

We prove that the partial $C^0$-estimate holds for metrics along Aubin's continuity method for finding Kähler-Einstein metrics, confirming a special case of a conjecture due to Tian. We use the method developed in recent work of Chen-Donaldson-Sun on the analogous problem for conical Kähler-Einstein metrics.

Paper proves a spinorial version of Aubin's estimate for the Yamabe problem.

New metrics found with specific curvature properties on 4D manifolds.

We characterize the global maximizers of a certain non-local functional defined on the space of all positively curved metrics on an ample line bundle L over a Kahler manifold X. This functional is an adjoint version, introduced by Berndtsson, of Donaldson's L-functional and generalizes the Ding-Tian functional whose cr…

Let X be a compact complex manifold equipped with a smooth (but not necessarily positive) closed form theta of one-one type. By a well-known envelope construction this data determines a canonical theta-psh function u which is not two times differentiable, in general. We introduce a family of regularizations of u, param…

The paper improves isoperimetric inequalities in noncompact manifolds with scalar curvature constraints.

New inequality on sphere generalizes circle inequality.

New insights into possible Gauss curvatures on $S^{2}$.

Study deforms Hermitian metrics with positive curvature.

On a n-dimensional connected compact manifold with non-empty boundary equipped with a Riemannian metric, a spin structure and a chirality operator, we study some properties of a spin conformal invariant defined from the first eigenvalue of the Dirac operator under the chiral bag boundary condition. More precisely, we s…

In this paper, we obtain the sharp $k$-th order Sobolev inequalities in the hyperbolic space ${\H}^n$ for all $k=1,2,3,\cdots$. This gives an answer to an open question raised by Aubin in [5, p.$\;$176-177] for $W^{k,2}({\H}^n)$ with $k>1$. In addition, we prove that the associated Sobolev constants are optimal.

Unified Schwarz lemma in Kähler and Hermitian geometry.

New metrics found without topological restrictions.

New proof of Sobolev inequality with constraints on sphere.

In this paper, we establish that: Suppose a closed Riemannian manifold $(M^n,g_0)$ of dimension $\geq 8$ is not locally conformally flat, then the Paneitz-Sobolev constant of $M^n$ has the property that $q(g_0)<q(S^n)$. The analogy of this result was obtained by T. Aubin in 1976 and had been used to solve the Yamabe pr…

Let ${\bf M}$ be a compact Riemannian manifold and the metrics $g=g(t)$ evolve by the Ricci flow. We prove the following result. The Sobolev imbedding by Aubin or Hebey, perturbed by a scalar curvature term and modulo sharpness of constants, holds uniformly for $({\bf M}, g(t))$ for all time if the Ricci flow exists fo…

On a Fano manifold M we study the supremum of the possible t such that there is a Kähler metric in c_1(M) with Ricci curvature bounded below by t. This is shown to be the same as the maximum existence time of Aubin's continuity path for finding Kähler-Einstein metrics. We show that on P^2 blown up in one point this sup…

New approach solves Calabi problem on manifolds with edge-cone singularities.