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Fagui Li

Publications and source records attributed to Fagui Li.

At least 19 recordsLinked to original sources
arXiv2026

Conformal Boundary Deformations under Ricci Lower Bounds: Eigenvalue Counterexamples

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.

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arXiv2026

Volume gap for minimal submanifolds in spheres, II

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 $\frac{(n+3)^{n+2}}{4(n+1)^n}\frac{\Vol(M)}{\Vol(\Sph^n)}$.

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arXiv2026

A Curvature Gap for Minimal Submanifolds in Spheres

Let $F:M^n\to\Sn^{n+q}(1)$ be a closed minimal immersion in the unit sphere, with $n\ge3$, $q\ge2$. Let $S$ be the squared length of its second fundamental form. We prove that if $M$ is not totally geodesic, then $$ \max_M S> \frac{n(288\sqrt6\,n-288)}{(288\sqrt6+192)n-137} >\frac{2n}{3}+\frac{31-9\sqrt6}{75}(n-2). $$

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arXiv2026

Pinching rigidity of surfaces with parallel mean curvature vector in spheres

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.

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arXiv2026

Lu's conjecture for minimal surfaces in codimension two

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.

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arXiv2026

On Chern's Conjecture for Minimal Submanifolds with Flat Normal Bundle in Spheres

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.

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arXiv2026

The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow

In this paper, we prove a spectral upper-pinching theorem for complete properly immersed self-shrinking hypersurfaces. Our argument is inspired by the second author's recent work\cite{Zhao2025}. If \(\lambda_\rho(\Sigma)\geq\lambda>0\) and \(S=|A|^2<1+\lambda\), then \(\Sigma\) is either a hyperplane or a generalized round cylinder. In the properly embedded case, the Ding--Xin and Brendle--Tsiamis weighted Poincar\'e estimate gives \(\lambda_\rho(\Sigma)\geq1/2\). Consequently, the pointwise upper pinching \(S<3/2\) forces \(\Sigma\) to be a hyperplane or a generalized round cylinder. For embedded self-shrinking surfaces in \(\mathbb R^3\), we also obtain the endpoint case \(S\leq3/2\). These results remove the lower pointwise pinching assumption in the corresponding embedded upper-pinching range and improve the ranges in earlier work of Ding--Xin~\cite{DingXin2014}, Cheng--Wei~\cite{ChengWei2015}, and Lei--Xu--Xu~\cite{LeiXuXu2020}.

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arXiv2026

Flat minimal tori and Lu's second-gap conjecture

Lu conjectured that, for each dimension and codimension, there exists a positive gap above the first pinching value for the quantity $S+\lambda_2$ on closed minimal submanifolds of the unit sphere, where $S$ is the squared norm of the second fundamental form and $\lambda_2$ is the second eigenvalue of Lu's fundamental matrix. We disprove this second-gap conjecture for minimal surfaces in every codimension $q\ge3$. More precisely, in every odd codimension $q\ge3$ we construct linearly full closed embedded flat minimal tori with $S\equiv2$ for which the constant values of $S+\lambda_2$ are dense in $(2,3)$. Thus the first pinching value $2$ can be approached from above by closed embedded minimal surfaces, and no uniform second gap exists in codimension at least three. Together with the known positive results in codimensions one and two, our examples complete the codimension picture for Lu's second-gap problem for minimal surfaces: the conjecture holds precisely in codimensions $1$ and $2$,

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arXiv2026

Gaussian-Weighted Curvature Gaps for Self-Shrinkers

In this paper, we prove lower bounds for the Gaussian-weighted \(L^2\)-curvature integral of embedded self-shrinkers. The proof combines normal coordinate functions with weighted Poincar\'e inequalities arising from first-eigenvalue estimates of Ding--Xin and Brendle--Tsiamis. For closed self-shrinkers, the estimate gives an explicit lower bound in terms of entropy and, together with the entropy gap theorem of Colding--Ilmanen--Minicozzi--White, yields a strict curvature gap for nonspherical closed shrinkers. In dimension two, we combine this estimate with the classification theorem and entropy gap theorem of Bernstein--Wang to obtain the corresponding gap statement for complete embedded self-shrinkers with polynomial volume growth.

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arXiv2026

Pinching rigidity theorems for normal scalar curvature

Let $M^n$ be an $n$-dimensional closed minimal submanifold immersed in the unit sphere $\mathbb{S}^{n+m}$. Denote by $S$ and $\rho^{\perp}$ the squared norm of the second fundamental form and the normal scalar curvature of $M^n$, respectively. Let $\{A^{\alpha}\}_{\alpha=n+1}^{n+m}$ be the shape operators of $M^n$ with respect to a local orthonormal normal frame. Denote by $\lambda_{1}$ the largest eigenvalue of the positive semi-definite symmetric matrix $\mathcal{A}=(\langle A^{\alpha},A^{\beta}\rangle)_{m\times m}$. We show that if $\lambda_{1}\leqslant n$ and $\rho^{\perp}\leqslant \left[{\sqrt{2}n(n-1)}\right]^{-1} \mathop{\inf}\limits_{p\in M}(n-\lambda_{1})(p)$, then $\rho^{\perp}\equiv 0$, which means the normal bundle of $M^n$ is flat, and further we give the classification of $M^n$.

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arXiv2026

On Simon's third gap conjecture for minimal surfaces in spheres

In this paper, continuing our previous work, we investigate the third gap problem in the Simon conjecture for closed minimal surfaces in the unit sphere. By developing refined third-order Simons-type integral identities and establishing new lower bounds for higher-order curvature terms, we obtain positive gap results throughout the entire interval $\left[\frac{5}{3},\frac{9}{5}\right]$ for the squared norm of the second fundamental form, including the endpoint cases. As an application, we establish a rigidity result for closed self-shrinkers.

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arXiv2026

Lu's conjecture for minimal surfaces

After Chern's conjecture on the discreteness of the constant scalar curvatures of compact minimal submanifolds $M^n$ in unit spheres $\mathbb{S}^{n+q}$, Z. Q. Lu proposed a conjecture regarding the second gap, based on his ingenious refinement of the known first gap theorem. This refinement unifies Simons' first gap theorem for hypersurfaces with the corresponding theorems for high-codimensional submanifolds established by Yau, Shen, Li and Li, among others. In this paper, for arbitrary codimension, we prove Lu's conjecture for minimal 2-spheres, and for any minimal surfaces under some slight inequality conditions about the normal scalar curvature.

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arXiv2023

A first eigenvalue estimate for embedded hypersurfaces in positive Ricci curvature manifolds

Let $\Sigma$ be a closed, embedded, oriented hypersurface in a closed oriented Riemannian manifold $N$. Under a lower bound on the Ricci curvature and an upper bound on the sectional curvature of $N$, we establish a lower bound for the first nonzero eigenvalue of the Laplacian on $\Sigma$. The estimate depends on the ambient curvature bounds, the normal injectivity radius, and the geometry of $\Sigma$ through its mean curvature and second fundamental form. This result extends the classical eigenvalue estimate of Choi and Wang [J. Diff. Geom. \textbf{18} (1983), 559--562.] to the non-minimal case.

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arXiv2023

An isoperimetric inequality of minimal hypersurfaces in spheres

Let $ M^n$ be a closed immersed minimal hypersurface in the unit sphere $\mathbb{S}^{n+1}$. We establish a special isoperimetric inequality of $M^n$. As an application, if the scalar curvature of $ M^n$ is constant, then we get a uniform lower bound independent of $M^n$ for the isoperimetric inequality. In addition, we obtain an inequality between Cheeger's isoperimetric constant and the volume of the nodal set of the height function.

math.DG↗
arXiv2022

Volume gap for minimal submanifolds in spheres

For a closed minimal submanifold $f:M^n\looparrowright \mathbb{S}^{N}$ in the unit sphere $(n<N)$, we prove $${\rm Vol}(M^n) \geq\frac{n+1}{n+2}\int_{M}\left( 1+\varphi_{p}^2\right) \geq m{\rm Vol}(\mathbb{S}^{n}),$$ where $\varphi_{p}(x):=\langle f(x),p\rangle$ is the height function in direction $p\in f(M)$, $m$ denotes the multiplicity of $p\in f(M)$ and ${\rm Vol}$ denotes the Riemannian volume functional, and each equality holds if and only if $M$ is totally geodesic. As an application, if the volume of $M^n$ is less than or equal to the volume of any $n$-dimensional minimal Clifford torus, then $M^n$ must be embedded, verifying the non-embedded case of Yau's conjecture. In addition, we also get volume gaps for minimal hypersurfaces with constant scalar curvature, improving Cheng-Li-Yau's classical volume gap in this case. Some other volume gaps and related pinching rigidities are also obtained.

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