On a multi-particle Moser-Trudinger Inequality

Trudinger-Moser embedding on the hyperbolic space

In [16], we established Trudinger-Moser inequalities for complete noncompact Riemannian manifold on which the Ricci curvature has lower bound and the injectivity radius is strictly positive. In this note, we improve those inequalties when the manifold is the hyperbolic space. The method we used here is still gluing local estimates.

A singular Moser-Trudinger inequality for mean value zero functions in dimension two

Let $Ω\subset\mathbb{R}^2$ be a smooth bounded domain with $0\in\partialΩ$. In this paper, we prove that for any $β\in(0,1)$, the supremum $$\sup_{u\in W^{1,2}(Ω), \int_Ωu dx=0, \int_Ω|\nabla u|^2dx\leq1}\int_Ω\frac{e^{2π(1-β) u^2}}{|x|^{2β}}dx$$ is finite and can be attained. This partially generalizes a well-known work of Alice Chang and Paul Yang (J. Differential Geom. 27 (1988), no. 2, 259-296) who have obtained the inequality when $β=0$.

关于曲面的Moser-Trudinger不等式

On the Moser-Trudinger inequality for surfaces

本文研究光滑黎曼曲面上的Moser-Trudinger不等式，证明相关常数可仅由systole、等周常数和曲率三个参数决定，并应用于构造双曲闭曲面的一致覆盖序列。

The Moser-Trudinger inequality on Kahler-Einstein manifolds

We prove the conjecture of Tian on the strong form of the Moser-Trudinger inequality for Kahler-Einstein manifolds with positive first Chern class, when there are no holomorphic vector fields, and, more generally, when the setting is invariant under a maximal compact subgroup of the automorphism group.

A generalized Trudinger-Moser inequality on a compact Riemannian surface

Let $(Σ, g)$ be a compact Riemannian surface. Let $ψ$, $h$ be two smooth functions on $Σ$ with $\int_Σψdv_g \neq 0$ and $h\geq0$, $h\not\equiv0$. In this paper, using a method of blowup analysis, we prove that the functional \begin{align}\label{functional_J} J^{ψ,h}(u)=\frac{1}{2}\int _Σ|\nabla_g u|^2dv_g + 8π\frac{1}{\int_Σψdv_g}\int_Σψudv_g-8π\log\int _Σhe^{u}dv_g \end{align} is bounded from below in $W^{1,2}(Σ,g)$. Moreover, we obtain a sufficient condition under which $J^{ψ, h}$ attains its infimum in $W^{1,2}(Σ,g)$. These results generalize the main results in \cite{DJLW97} and \cite{YZ20…

双曲空间上无边界条件对称函数的精确Sobolev与Adams-Trudinger-Moser不等式

Sharp Sobolev and Adams-Trudinger-Moser inequalities for symmetric functions without boundary conditions on hyperbolic spaces

本文针对双曲空间中无零边界条件的径向函数，建立了精确Sobolev与Adams-Trudinger-Moser嵌入定理，提出高阶协变导数与径向导数范数的非平凡比较结果，补充了双曲空间无边界对称函数嵌入理论的空白。

Improved Moser-Trudinger-Onofri inequality under constraints

A classical result of Aubin states that the constant in Moser-Trudinger-Onofri inequality on $\mathbb{S}^{2}$ can be imporved for furnctions with zero first order moments of the area element. We generalize it to higher order moments case. These new inequalities bear similarity to a sequence of Lebedev-Milin type inequalities on $\mathbb{S}^{1}$ coming from the work of Grenander-Szego on Toeplitz determinants (as pointed out by Widom). We also discuss the related sharp inequality by a perturbation method.

Trudinger-Moser inequalities on a closed Riemannian surface with the action of a finite isometric group

Let $(Σ,g)$ be a closed Riemannian surface, $W^{1,2}(Σ,g)$ be the usual Sobolev space, $\textbf{G}$ be a finite isometric group acting on $(Σ,g)$, and $\mathscr{H}_\textbf{G}$ be a function space including all functions $u\in W^{1,2}(Σ,g)$ with $\int_Σudv_g=0$ and $u(σ(x))=u(x)$ for all $σ\in \textbf{G}$ and all $x\inΣ$. Denote the number of distinct points of the set $\{σ(x): σ\in \textbf{G}\}$ by $I(x)$ and $\ell=\inf_{x\in Σ}I(x)$. Let $λ_1^\textbf{G}$ be the first eigenvalue of the Laplace-Beltrami operator on the space $\mathscr{H}_\textbf{G}$. Using blow-up analysis, we prove that if $α<…

On energy functionals, Kahler-Einstein metrics, and the Moser-Trudinger-Onofri neighborhood

We prove that the existence of a Kahler-Einstein metric on a Fano manifold is equivalent to the properness of the energy functionals defined by Bando, Chen, Ding, Mabuchi and Tian on the set of Kahler metrics with positive Ricci curvature. We also prove that these energy functionals are bounded from below on this set if and only if one of them is. This answers two questions raised by X.-X. Chen. As an application, we obtain a new proof of the classical Moser-Trudinger-Onofri inequality on the two-sphere, as well as describe a canonical enlargement of the space of Kahler potentials on which thi…

Moser-Trudinger type inequalities for complex Monge-Ampère operators and Aubin's "hypothèse fondamentale"

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 the precise value proposed by Aubin). In the different setting of pseudoconvex domains in complex space we also obtain a quasi-sharp version of the inequalities and relate it to Brezis-Merle type inequalities. The inequalities are shown to be sharp for S^{1}-invariant functions …

A thermodynamical formalism for Monge-Ampere equations, Moser-Trudinger inequalities and Kahler-Einstein metrics

We develop a variational calculus for a certain free energy functional on the space of all probability measures on a Kahler manifold X. This functional can be seen as a generalization of Mabuchi's K-energy functional and its twisted versions to more singular situations. Applications to Monge-Ampère equations of mean field type, twisted Kahler-Einstein metrics and Moser-Trudinger type inequalities on Kahler manifolds are given. Tian's α- invariant is generalized to singular measures, allowing i particular the construction of Kahler-Einstein metrics with positive Ricci curvature that are singula…

黎曼流形上的相对法伯-克拉恩不等式与特鲁丁格方程

Relative Faber--Krahn inequalities and Trudinger's equation on Riemannian Manifolds

该研究在黎曼流形上探讨特鲁丁格方程，证明相对p-法伯-克拉恩不等式等价于体积加倍与次高斯上估计的结合，还导出了改进的长时间上估计。

用径向空间支配流形

Dominating manifolds by radial spaces

本文通过构造 CD(k,n)-空间上满足 Polya-Szegö 型不等式的函数重排，证明了一大类度量测度空间上的 Moser-Trudinger 型不等式，并刻画了 Ricci 曲率有下界流形中容许该不等式的对象。