Given a klt singularity $x\in (X, D)$, we show that a quasi-monomial valuation v with a finitely generated associated graded ring is a minimizer of the normalized volume function ${\widehat{\text{vol}}}_{(X,D),x}$, if and only if v induces a degeneration to a K-semistable log Fano cone singularity. Moreover, such a minimizer is unique among all quasi-monomial valuations up to rescaling. As a consequence, we prove that for a klt singularity $x\in X$ on the Gromov–Hausdorff limit of Kähler–Einstein Fano manifolds, the intermediate K-semistable cone associated with its metric tangent cone is uniquely determined by the algebraic structure of $x\in X$, hence confirming a conjecture by Donaldson–Sun.
In this paper, we prove the existence of solutions to the Fu–Yau equation on compact Kähler manifolds. As an application, we give a class of non-trivial solutions of the modified Strominger system.
This is the first part of a series of papers on the spectrum of the SYK model, which is a simple model of the black hole in physics literature. In this paper, we will give a rigorous proof of the almost sure convergence of the global density of the eigenvalues. We also discuss the largest eigenvalue of the SYK model.
In this paper, we study the isotropic–nematic phase transition for the nematic liquid crystal based on the Landau–de Gennes $\mathbf {Q}$-tensor theory. We justify the limit from the Landau–de Gennes flow to a sharp interface model: in the isotropic region, $\mathbf {Q}\equiv 0$; in the nematic region, the $\mathbf {Q}$-tensor is constrained on the manifolds $\mathcal {N}=\{s_+(\mathbf {n}\otimes \mathbf {n}-\frac{1}{3}\mathbf {I}), \mathbf {n}\in {\mathbb {S}^2}\}$ with $s_+$ a positive constant, and the evolution of alignment vector field $\mathbf {n}$ obeys the harmonic map heat flow, while the interface separating the isotropic and nematic regions evolves by the mean curvature flow. This problem can be viewed as a concrete but representative case of the Rubinstein–Sternberg–Keller problem introduced in Rubinstein et al. (SIAM J. Appl. Math. 49:116–133,
We define a formal Gromov–Witten theory of the quintic threefold via localization on ${\mathbb {P}}^4$. Our main result is a direct geometric proof of holomorphic anomaly equations for the formal quintic in precisely the same form as predicted by B-model physics for the true Gromov–Witten theory of the quintic threefold. The results suggest that the formal quintic and the true quintic theories should be related by transformations which respect the holomorphic anomaly equations. Such a relationship has been recently found by Q. Chen, S. Guo, F. Janda, and Y. Ruan via the geometry of new moduli spaces.
We study the flatness of log-pluricanonical sheaves on stable families of varieties.
We study the flatness of log-pluricanonical sheaves on stable families of surfaces.
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 Chang and Wang. Our proof uses an alternative and elementary argument.