A multiplicity estimate for the Jacobi operator of a nonflat Yang-Mills field over $\mathbb{S}^m$

Jacobi operators along the structure flow on real hypersurfaces in a nonflat complex space form

Let $M$ be a real hypersurface of a complex space form with almost contact metric structure $(ϕ, ξ, η, g)$. In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator $R_ξ=R(\cdot,ξ)ξ$ is $ξ$-parallel. In particular, we prove that the condition $\nabla_ξ R_ξ=0$ characterizes the homogeneous real hypersurfaces of type $A$ in a complex projective space or a complex hyperbolic space when $R_ξ ϕS=S ϕR_ξ$ holds on $M$, where $S$ denotes the Ricci tensor of type (1,1) on $M$.

$G_2$-瞬子于 $S^7$ 上的Morse指标

Morse Index of a $G_2$-Instanton over $S^7$

本文证明四元数Hopf纤维化拉回的BPST瞬子在$S^7$上达到Morse指标下界8，零化度20，其Yang--Mills核包含Waldron的15维形变空间及5个额外零模。

Compactness and generic finiteness for free boundary minimal hypersurfaces (I)

Given a compact Riemannian manifold with boundary, we prove that the space of embedded, which may be improper, free boundary minimal hypersurfaces with uniform area and Morse index upper bound is compact in the sense of smoothly graphical convergence away from finitely many points. We show that the limit of a sequence of such hypersurfaces always inherits a non-trivial Jacobi field when it has multiplicity one. In a forthcoming paper, we will construct Jacobi fields when the convergence has higher multiplicity.

Spectrum of the Laplacian and the Jacobi operator on rotational cmc hypersurfaces of spheres

Let $M\subset \mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}$ be a compact cmc rotational hypersurface of the $(n+1)$-dimensional Euclidean unit sphere. Denote by $|A|^2$ the square of the norm of the second fundamental form and $J(f)=-Δf-nf-|A|^2f$ the stability or Jacobi operator. In this paper we compute the spectra of their Laplace and Jacobi operators in terms of eigenvalues of second order Hill's equations. For the minimal rotational examples, we prove that the stability index --the numbers of negative eigenvalues of the Jacobi operator counted with multiplicity -- is greater than $3 n+4$ and w…

Contact Isotropic Realisations of Jacobi Manifolds via Spencer Operators

Motivated by the importance of symplectic isotropic realisations in the study of Poisson manifolds, this paper investigates the local and global theory of contact isotropic realisations of Jacobi manifolds, which are those of minimal dimension. These arise naturally when considering multiplicity-free actions in contact geometry, as shown in this paper. The main results concern a classification of these realisations up to a suitable notion of isomorphism, as well as establishing a relation between the existence of symplectic and contact isotropic realisations for Poisson manifolds. The main too…

弹跳雅可比场与曲面上的 Allen-Cahn 方程

Bouncing Jacobi fields and the Allen-Cahn equation on surfaces

本文在紧致曲面上构造了 Allen-Cahn 泛函的可数族临界点，其节点集以重数 2 收敛到给定嵌入测地线，且能量与 Morse 指标一致有界，并指出该现象为二维曲面特有。