Inspired by the Simon conjecture for minimal surfaces in spheres, we study closed surfaces with parallel mean curvature vector and positive Gaussian curvature immersed in unit spheres. Let $h$ be the second fundamental form, let $\mathbf{H}$ be the mean curvature vector field, and set $\tilde h=h-\mathbf{H}g$ and $\tilde S=|\tilde h|^2=|h|^2-2H^2$, where $H=|\mathbf{H}|$ and $g$ is the induced Riemannian metric on the surface $M$. We establish three Simons-type integral identities for $\tilde S$, which extend the first, second and third gap identities in the minimal case. As applications, we obtain the first two sharp endpoint gaps and several rigidity and oscillation estimates in the third interval. We further characterize the endpoint cases by combining these identities with the classification theorems of Calabi and Yau.
Let $M^n$ $(n\geqslant3)$ be a closed minimal submanifold in the unit sphere $\mathbb S^{n+m}$ $(m\geqslant2)$ with flat normal bundle, and let $S$ denote the squared norm of its second fundamental form. We prove an explicit second-gap rigidity theorem for $S$. More precisely, if $S$ is constant and \[ 0\leqslant S\leqslant n+\delta, \] where $\delta$ is an explicit constant satisfying $\delta\geqslant \frac{n}{87}$, then either $S\equiv0$ and $M$ is a totally geodesic sphere, or $S\equiv n$ and $M$ is a Clifford torus contained in a totally geodesic $\mathbb S^{n+1}\subset\mathbb S^{n+m}$. %We observe that the flat-normal-bundle assumption is necessary here. The flat-normal-bundle condition is essential in the general higher-codimensional setting: without it, the corresponding rigidity statement already fails in dimension two. This theorem provides positive evidence for Chern's conjecture in higher codimension.
Let $f:M^n\looparrowright\Sph^{n+q}(1)$, $n\ge2$ and $q\ge1$, be a closed, connected, non-totally-geodesic minimal immersion with second fundamental form $h$, and put $S=|h|^2$ and $S_*=\max_M S$. If $p\in f(M)$ has multiplicity $m$ and $f^{-1}(p)=\{x_1,\ldots,x_m\}$, then \[ \Vol(M)\ge \left[m+\varepsilon_n\sum_{j=1}^m \left(\frac{S(x_j)}{S_*}\right)^2\right]\Vol(\Sph^n), \] where $[110n(n+2)^2]^{-1}<\varepsilon_n<[104n(n+2)^2]^{-1}$. If the immersion is linearly full, then \[ \frac{\Vol(M)}{\Vol(\Sph^n)} \ge \max\!\left\{1+\varepsilon_n, \frac{4(n+1)^n}{(n+3)^{n+2}}(n+q+1)\right\}. \] Moreover, for every hyperplane $H$ through the origin, each connected component of $M\setminus f^{-1}(H)$ has volume at least $4(n+1)^n(n+3)^{-n-2}\Vol(\Sph^n)$; consequently the number of components is at most $\frac{(n+3)^{n+2}}{4(n+1)^n}\frac{\Vol(M)}{\Vol(\Sph^n)}$.
Let $M^2\to\mathbb{S}^4$ be a closed minimal immersion, let $S$ be the squared norm of its second fundamental form, and let $\lambda_1\geq\lambda_2\geq0$ be the eigenvalues of Lu's fundamental matrix. We classify all such immersions for which $S+\lambda_2$ is constant. We prove that the constant can only be $0$ or $2$. In the first case the image is a totally geodesic $2$-sphere; in the second case it is either a Clifford torus in a totally geodesic $\mathbb{S}^3$ or the Veronese surface in $\mathbb{S}^4$. In particular, there is no closed minimal surface in $\mathbb{S}^4$ with constant $S+\lambda_2>2$. Consequently, Lu's second-gap conjecture holds for minimal surfaces in codimension two. Together with the hypersurface result of Peng--Terng and the counterexamples of Li--Zhao in every codimension $m\geq3$, this completes the codimension picture for minimal surfaces.
Let $F:M^n\to\Sn^{n+q}(1)$ be a closed connected minimal immersion in the unit sphere with second fundamental form $h$, $n\ge3$, $q\ge2$, and $S=|h|^2$. We prove that if $M$ is not totally geodesic, then \[ \max_M S\ge \frac{2n}{3}+\frac{n-2}{6300(39n+8)} \ge\frac{2n}{3}+\frac1{787500}. \]
Let $(M^{n+1},g)$ be a compact Riemannian manifold with boundary. Under the assumptions $\Ric_g\geq ng$ and $\II_g\geq0$, Wang proposed a sharp strengthening of the Choi--Wang--Reilly estimate, asserting that the first nonzero Laplace eigenvalue of the boundary is at least $n$; see [J. Geom. Anal. 31 (2021)]. We disprove this assertion in every dimension $n+1\geq3$. More precisely, we construct a sequence of metrics on the hemisphere $\Sph^{n+1}_{+}$ converging in $C^\infty$ to the round metric and satisfying \[ \Ric_g>n g,\qquad \II_g>0,\qquad \lambda_1(\partial\Sph^{n+1}_{+},g|_{\partial\Sph^{n+1}_{+}})<n. \] The construction starts from Zhu's infinitesimal conformal deformation, which lowers one branch of the first boundary eigenspace while preserving the normalized Ricci lower bound to first order. We add a multiple of the spherical height function.
Fagui Li, Yu-Hang Zhao· 0 citations
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