For a smooth immersion $F:\mathbb{R}\mathbb{P}^m\looparrowright \overline{\mathbb{B}}^{N}(1)$, we observe that the sharp systolic inequality forces a sharp lower bound on its maximal normal curvature $\kappa(F)$. In dimensions $m=2,3$, the sharp inequalities of Pu and Bray--Brendle--Eichmair--Neves therefore give $\kappa(F)^2\ge \frac{2m}{m+1}$. Equality holds precisely for the Veronese embedding. This recovers Petrunin's theorem for $\mathbb{R}\mathbb{P}^2$ and, for $\mathbb{R}\mathbb{P}^3$, confirms the first open case of his question for real projective spaces.
Let $M^4\hookrightarrow\mathbb S^5(1)$ be a closed CMC hypersurface with constant scalar curvature and constant third power sum $f_3=\sum_{i,j,k}h_{ij}h_{jk}h_{ki}$. We prove that if $M^4$ has exactly two distinct principal curvatures at some point, then it is isoparametric. More precisely, it is a Clifford torus of th...
For any oriented smooth $4$--manifold $X$ diffeomorphic to $(S^2\times D^2)^{\natural n}$ ($n\geq0$), the author establishes a natural isomorphism of abelian groups: $$\mathrm{Mod}(X,\partial X)\cong \mathrm{Mod}(D^4,\partial D^4)\times\wedge^2H_2(X;\mathbb{Z}),$$ concerning the (smooth) boundary-fixing mapping class g...
Let $(M^3, g_{\mathrm{hyp}})$ be a closed oriented hyperbolic three-manifold normalized so that $\operatorname{sec}_{g_{\mathrm{hyp}}} \equiv -1$. We prove a sharp lower bound for the volume of hypersurfaces in $M^3 \times \mathbb{S}^1$ representing the slice class $[M^3 \times \{ \mathrm{pt} \}] \in H_3(M^3 \times \ma...
Lawson showed that a non-totally geodesic Einstein minimal hypersurface in $\mathbb S^5$ is congruent to the Clifford hypersurface $\mathbb S^2(1/\sqrt2)\times \mathbb S^2(1/\sqrt2).$ It is also known, by work of Cartan and \^{O}tsuki, that a non-totally geodesic locally conformally flat minimal hypersurface in $\mathb...
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 $f:M^n\looparrowright\Sph^{n+q}(1)$, $n\ge2$ 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}^...
Jian-Quan Ge, Fa-Gui Li· 1 citation
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