arXiv research

The paper generalizes inequalities on almost Kähler manifolds.

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 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…

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

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

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…

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

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…

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

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

The Yamabe problem in compact closed Riemannian manifolds is concerned with finding a metric with constant scalar curvature in the conformal class of a given metric. This problem was solved by the combined work of Yamabe, Trudinger, Aubin, and Schoen. In particular, Aubin solved the case when the Riemannian manifold is…

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…

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

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

Reverse Hölder inequalities on Fano metrics with applications to geodesics and singularities.

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

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

Solves complex equation for specific geometric solitons.

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…

Derives scalar reduction for generalized Kähler-Ricci solitons, proving uniqueness.

The paper studies the twisted Calabi flow on Kähler manifolds.

We study the free energy of the Laughlin state on curved backgrounds, starting from the free field representation. A simple argument, based on the computation of the gravitational effective action from the transformation properties of Green functions under the change of the metric, allows to compute the first three ter…

Zeta functions for non-unitary twists are shown to have analytic continuation.

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.

Survey on twisted dynamical zeta functions and Fried's conjecture.

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

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

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 inequality on sphere generalizes circle inequality.

On four-dimensional closed manifolds we introduce a class of canonical Riemannian metrics, that we call weak harmonic Weyl metrics, defined as critical points in the conformal class of a quadratic functional involving the norm of the divergence of the Weyl tensor. This class includes Einstein and, more in general, harm…

In this paper we study the Palais-Smale sequences of the conformal Dirac-Einstein problem. After we characterize the bubbling phenomena, we prove an Aubin type result leading to the existence of a positive solution. Then we show the existence of infinitely many solutions to the problem provided that the underlying mani…

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…

Let $(M,F)$ be a $C^\infty$ Finsler manifold, $p\geq 1$ a real number, $k$ a positive integer and $H_k^p (M)$ a certain Sobolev space determined by a Finsler structure $F$. Here, it is shown that the set of all real $C^{\infty}$ functions with compact support on $M$ is dense in the Sobolev space $H_1^p (M)$. This resul…

Reconstruct Lie structures from functional-analytic data on groupoids.

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.

Given a closed symplectic 4-manifold $(X,ω)$, we define a twisted version of the Gromov-Taubes invariants for $(X,ω)$, where the twisting coefficients are induced by the choice of a surface bundle over $X$. Given a fibered 3-manifold $Y$, we similarly construct twisted Lefschetz zeta functions associated with surface b…

Given a three dimensional pseudo-Einstein CR manifold $(M,T^{1,0}M,θ)$, we study the existence of a contact structure conformal to $θ$ for which the logarithmic Hardy-Littlewood-Sobolev (LHLS) inequality holds. Our approach closely follows \cite{Ok1} in the Riemannian setting. For this purpose, we introduce the notion …

Study of twisted Ruelle zeta function on hyperbolic manifolds and its relation to analytic torsion.

We study the existence of solutions of the non-linear differential equations on the compact Riemannian manifolds $(M^n,g), n\geq 2$, Δ_p u + a(x)u^{p-1} = λf(u,x), (E2) where $Δ_p$ is the $p-$laplacian, with $1<p<n$. The equation (E2) generalizes a equation considered by Aubin, where he has considered, a compact Rieman…

Unified Schwarz lemma in Kähler and Hermitian geometry.

New metrics found without topological restrictions.

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…

New proof of Sobolev inequality with constraints on sphere.

Analogous zeta function for twisted Alexander invariants defined.

Let $X\hookrightarrow \cpn $ be a smooth complex projective variety of dimension $n$. Let $λ$ be an algebraic one parameter subgroup of $G:=\gc$. Let $0\leq l\leq n+1$. We associate to the coefficients $F_{l}(λ)$ of the normalized weight of $λ$ on the $mth$ Hilbert point of $X$ new energies $F_{\om,l}(\vp)$. The (loga…