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 $[M^3 \times \{ \mathrm{pt} \}] \in H_3(M^3 \times \mathbb{S}^1; \mathbb{Z})$, and we classify the equality case. If $g$ is a smooth Riemannian metric on $M^3 \times \mathbb{S}^1$ with the scalar curvature $\operatorname{Sc}_g \geq -6$, then every closed embedded hypersurface $\Sigma$ representing the slice class $[M^3\times\{\mathrm{pt}\}]$ satisfies $\operatorname{vol}_g(\Sigma) \geq \operatorname{vol}_{g_{\mathrm{hyp}}}(M^3)$. The bound is attained by the product metric $g_{\mathrm{hyp}}+h$, with $h$ any metric on $\mathbb{S}^1$. Conversely, if equality holds for some $\Sigma$, then up to a diffeomorphism preserving the slice class, $g=g_{\mathrm{hyp}}+h$ and $\Sigma = M^3 \times \{\mathrm{pt}\}$.
We find an explicit rank-one convex non-quasiconvex integrand in $\mathbb{R}^{2\times 4}$: to falsify the quasiconvexity inequality, we exhibit a map $\mathbb{T}^4\to \mathbb{R}^2$ with $12$ non-zero Fourier modes. In fact, this map is obtained from a scalar potential, so we also find a rank one convex integrand in $\m...
Andrea Agazzi, Giuseppe Bruno, André Guerra et al.· 0 citations
Let $u:\mathbb{S}^{2n+1}\to\mathbb{CP}^n$ be a smooth nonconstant harmonic map. We prove that its Morse index is equal to $ 2n+2$ if and only if \begin{equation*} u = h \circ \pi \circ \Xi, \end{equation*} where $\Xi:\mathbb{S}^{2n+1}\to\mathbb{S}^{2n+1}$ is an isometric transformation, $\pi:\mathbb{S}^{2n+1}\to\mathbb...
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...
We prove that for every even integer $N\geq 4$ or $N\in\{3,5,7\}$, every axially symmetric solution to the $Q$-curvature-type problem $$ \alpha P_N u + (N-1)!(1-\frac{e^{Nu}}{\int_{\mathbb{S}^N} e^{Nu}dw})=0 \ \ \ \ \ \mbox{on} \ \mathbb{S}^N $$ is constant, provided that $ \alpha\ge\frac{1}{2}$ and $\alpha \not =1$. T...
Chang-Feng Gui, Tuo Li, Jun-Cheng Wei et al.· 0 citations
We prove a sharp $2$-systolic inequality for four-dimensional products $\mathbb S^2\times P$, where $P\subset\mathbb R^2$ is an arbitrary convex polygon. Let $h=g_{\mathbb S^2}+g_{\mathrm{eu}}$ be the standard product metric. If a Riemannian metric $g$ on $\mathbb S^2\times P$ has scalar curvature $\geq \sigma>0$, nonn...
For $m=2,3$, we prove that every smooth immersion $F:\mathbb{R}\mathbb{P}^m\looparrowright\overline{\mathbb B}^{,N}(1)$ satisfies $\kappa(F)^2\ge 2m/(m+1)$, with equality only for the Veronese embedding, up to congruence. We also prove that a closed, connected, orientable three-manifold admitting an immersion into a Eu...
Tsz-Kiu Aaron Chow, Jingbo Wan· 0 citations
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