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Let $(M^n,g)$ be a compact Riemannian manifold with boundary $\partial M=N$. Suppose $|Rm_g| \le C_1$ on $M$ and the second fundamental form of $N$ is bounded by $C_2$. Moreover, there exists a consta …

Let $\Sigma$ be a closed surface smoothly embedded in $\mathbb R^3$. For any Morse function $h:\mathbb R^3 \to \mathbb R$, can we isotope $\Sigma$ so that the restriction of $h$ on $\Sigma$ is also a …

Let $S^3/\Gamma$ be a spherical space form where $\Gamma$ is a finite subgroup of $O(4)$ acting freely on $S^3$. If $\Gamma$ is trivial, it is well-known that the spectra of the Laplacian operator on …

Suppose $(M^n,g)$ is an $n$-dimensional Einstein manifold with $Ric=(n-1)g$. Let $\lambda$ be the minimal eigenvalue of the Lichnerowicz Laplacian $\Delta_L$ defined on all transverse-traceless symmet …

Let $(M^4,g)$ be a complete $4$-dimensional Riemannian manifold such that two almost complex structures $I$ and $J$ are compatible with $g$ and $\nabla_g I=\nabla_g J=0$. Can we prove that $(M,g)$ is …

Let $M$ be an oriented closed $3$-manifold equipped with an effective smooth circle action. Can we have a classification of all such $M$ such that there exists a $4$-manifold $N$ with $\partial N=M$, …