Problem 1 (yau-contest 2016)
Section: Geometry and Topology

Let $M$ be a compact odd-dimensional manifold with boundary $\partial M$. Show that the Euler characteristics of $M$ and $\partial M$ are related by:

$$\chi(M) = \frac{1}{2} \chi(\partial M).$$

Problem 2 (yau-contest 2016)
Section: Geometry and Topology

Compute the de Rham cohomology of a punctured two-dimensional torus $T^2 - \{p\}$, where $p \in T^2$. If $T^2 = \mathbb{R}^2/\mathbb{Z}^2$ with coordinates $(x, y) \in \mathbb{R}^2$, then is the volume form $\omega = dx \wedge dy$ exact?

Problem 3 (yau-contest 2016)
Section: Geometry and Topology

Let $M^n \to \mathbb{R}^{n+1}$ be a closed oriented hypersurface. The $r$-th mean curvature of $M^n$ is defined by

$$H_r := \frac{1}{\binom{n}{r}} \sum_{i_1 < i_2 < \cdots < i_r} \lambda_{i_1} \lambda_{i_2} \cdots \lambda_{i_r}, \quad (1 \le r \le n)$$

where $\lambda_i$ ($i=1,\cdots,n$) are principal curvatures of $M^n$. Prove that if all of $\lambda_i$ are positive and $H_r=$ constant for a certain $r$, then $M^n$ is a hypersphere in $\mathbb{R}^{n+1}$.

Problem 4 (yau-contest 2016)
Section: Geometry and Topology

State and prove the cut-off function lemma on a differentiable manifold.

Problem 5 (yau-contest 2016)
Section: Geometry and Topology

Let $M$ be a compact Riemannian manifold without boundary. Show that if $M$ has positive Ricci curvature, then $H^1(M,\mathbb{R}) = 0$.

Problem 6 (yau-contest 2016)
Section: Geometry and Topology

Let $M$ be an orientable, closed and embedded minimal hypersurface in $S^{n+1}$. Denote by $\lambda_1$ the first eigenvalue for the Laplace-Beltrami operator on $M$. Prove that $\lambda_1 \ge n/2$.