Topology of the nodal set of random equivariant spherical harmonics on $S^3$

A. Stern's analysis of the nodal sets of some families of spherical harmonics revisited

In this paper, we revisit the analyses of Antonie Stern (1925) and Hans Lewy (1977) devoted to the construction of spherical harmonics with two or three nodal domains. Our method yields sharp quantitative results and a better understanding of the occurrence of bifurcations in the families of nodal sets.This paper is a natural continuation of our critical reading of A. Stern's results for Dirichlet eigenfunctions in the square, see arXiv:14026054.

Tangent nodal sets for random spherical harmonics

In this note, we consider a fixed vector field $V$ on $S^2$ and study the distribution of points which lie on the nodal set (of a random spherical harmonic) where $V$ is also tangent. We show that the expected value of the corresponding counting function is asymptotic to the eigenvalue with a leading coefficient that is independent of the vector field $V$. This demonstrates, in some form, a universality for vector fields up to lower order terms.

Ruled nodal surfaces of Laplace eigenfunctions and injectivity sets for the spherical mean Radon transform in $\mathbb R^3.$

It is proved that if a Paley-Wiener family of eigenfunctions of the Laplace operator in $\mathbb R^3$ vanishes on a real analytically ruled two-dimensional surface $S \subset \mathbb R^3$ then $S$ is a union of cones, each of which is contained in a translate of the zero set of a nonzero harmonic homogeneous polynomial. If $S$ is an immersed $C^1-$ manifold then $S$ is a Coxeter system of planes. Full description of common nodal sets of the Laplace spectra of convexly supported distributions is given. In equivalent terms, the result describes ruled injectivity sets for the spherical mean trans…

极化球面流形的等变 $\mathbb R$-测试构型

Equivariant $\mathbb R$-test configurations of polarized spherical varieties

本文通过组合数据对极化球面流形的 $G$-等变正规 $\mathbb R$-测试构型进行了分类，证明了特殊构型中心纤维的有限性定理，并应用于研究 $\mathbb Q$-Fano 球面流形的半稳定退化问题。

等变极小最大理论与四维球面的球形伯恩斯坦问题

Equivariant min-max theory and the spherical Bernstein problem in $\mathbb{S}^4$

本文通过等变极小最大理论，在四维球面中构造了非赤道极小超球面，解决了Chern的球形伯恩斯坦问题，并展示了相关问题的解的正则性。

K-Stability of Fano spherical varieties

We prove a criterion for K-stability of a $\mathbb{Q}$-Fano spherical variety with respect to equivariant special test configurations, in terms of its moment polytope and some combinatorial data associated to the open orbit. Combined with the equivariant version of the Yau-Tian-Donaldson conjecture for Fano manifolds proved by Datar and Székelyhidi, it yields a criterion for the existence of a Kähler-Einstein metric on a spherical Fano manifold. The results hold also for modified K-stability and existence of Kähler-Ricci solitons.