D. H. Phong, Jian Song, Jacob Sturm, Ben Weinkove
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.
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Though Trudinger-Moser inequalities on compact Riemannian manifolds or Euclidean space are well understood, we know little about them on complete noncompact Riemannian manifolds. In this paper, we established respectively necessary condition and sufficient condition under which Trudinger-Moser inequalities hold on complete noncompact Riemannian manifolds. For applications of such inequalities, we obtained existence results for some quasilinear equations with nonlinearity of exponential growth.
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.
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。
Sharp Sobolev and Moser-Trudinger inequalities on noncompact Riemannian manifolds with Ricci curvature bounded below
本文研究了非紧致黎曼流形上具有下界曲率的Sobolev和Moser-Trudinger不等式，得出了最佳常数。
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…
We verify a conjecture of Gillet-Soulé. We prove that the determinant of the Laplacian on a line bundle over $\mathbb{CP}^{1}$ is always bounded from above. This can also be viewed as a multi-particle generalization of the Moser-Trudinger Inequality. Furthermore, we conjecture that this functional achieves its maximum at the canonical metric. We give some evidence for this conjecture, as well as links to other fields of analysis.
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.
We derive the sharp Moser-Trudinger-Onofri inequalities on the standard $n$-sphere and CR $(2n+1)$- sphere as the limit of the sharp fractional Sobolev inequalities for all $n\ge 1$. On the $2$-sphere and $4$-sphere, this was established recently by S.-Y. Chang and F. Wang. Our proof uses an alternative and elementary argument.
In the first part of the paper we investigate some geometric features of Moser-Trudinger inequalities on complete non-compact Riemannian manifolds. By exploring rearrangement arguments, isoperimetric estimates, and gluing local uniform estimates via Gromov's covering lemma, we provide a Coulhon, Saloff-Coste and Varopoulos type characterization concerning the validity of Moser-Trudinger inequalities on complete non-compact $n-$dimensional Riemannian manifolds $(n\geq 2)$ with Ricci curvature bounded from below. Some sharp consequences are also presented both for non-negatively and non-positive…
Constrained Moser-Trudinger-Onofri inequality and a uniqueness criterion for the mean field equation
本文建立了二阶矩偏离0约束下的Moser-Trudinger-Onofri不等式，并证明该偏离阈值是常数$a$接近$\frac{1}{2}$时平均场方程的唯一性准则。
Motivated by a recent work of X. Chen and M. Zhu (Commun. Math. Stat., 1 (2013) 369-385), we establish a Trudinger-Moser inequality on compact Riemannian surface without boundary. The proof is based on blow-up analysis together with Carleson-Chang's result (Bull. Sci. Math. 110 (1986) 113-127). This inequality is different from the classical one, which is due to L. Fontana (Comment. Math. Helv., 68 (1993) 415-454), since the Gaussian curvature is involved. As an application, we improve Chen-Zhu's result as follows: A modified Liouville energy of conformal Riemannian metric has a uniform lower …
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 …
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 $β=0$.
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…
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。
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…
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<…
Mabuchi introduced multiplier Hermitian structures on compact Kahler manifolds and defined metrics similar to Kahler-Einstein metrics under these structures. In this note we generalize the inequality of Moser-Trudinger type on Kahler-Einstein manifolds to this case.
We prove a conjecture saying that complex projective space has maximal volume (degree) among all toric Kaehler-Einstein manifolds of dimension n. The proof is inspired by our recent work on sharp Moser-Trudinger and Brezis-Merle type inequalities for the complex Monge-Ampere operator, but is essentially self-contained.
Through the study of some elliptic and parabolic fully nonlinear PDEs, we establish conformal versions of quermassintegral inequality, the Sobolev inequality and the Moser-Trudinger inequality for the geometric quantities associated to the Schouten tensor on locally conformally flat manifolds.
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。
In this article and in its sequel we propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamics on certain infinite-dimensional spaces. We illustrate the fruitfulness of this approach in the context of the Ricci flow, as well as another flow, in Kahler geometry. We introduce and study dynamical systems related to the Ricci operator on the space of Kahler metrics that arise as discretizations of these flow…
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