21 problems

Let $(M^{n+1},g)$ be a closed Riemannian manifold with $3\le n+1\le 7$. A closed hypersurface encloses half the volume of $M$ when one of the regions it bounds has volume equal to…

Equivariant Yau conjecture. The manifold $M$ contains infinitely many closed embedded minimal $G$-hypersurfaces $\Sigma$ in the given $G$-homology class. This is the equivariant ge…

Let $(M^{n+1},g)$ be a closed Riemannian manifold and let $[\Sigma_0]\in H_n(M^{n+1};\mathbb Z_2)$ be a homology class represented by a closed embedded hypersurface. Homology-class…

The $16\pi$ conjecture. The number $\beta_0$ is attained by a path $\omega\in C^0([0,1],\operatorname{Imm}(\mathbb{S}^2,\mathbb{R}^3))$ such that both restrictions…

Let $\Omega\subset \mathbb{R}^2$ be a bounded domain with smooth boundary, and let $p=1,2,\dots$. A billiard trajectory is a trajectory in $\Omega$ satisfying the billiard reflecti…

Marques–Neves' width stability conjecture. Then $(\mathbb S^3,g_k)$ converges in the Sormani–Wenger intrinsic flat sense to the round unit sphere $(\mathbb S^3,g_{rd})$. The conjec…

Let $g$ be a Riemannian metric and let a min-max sequence from a sweepout of genus $g$ surfaces converge as a varifold to … where the pairwise disjoint, smooth, closed, embedded mi…

Let $M$ be a non-Haken $3$-manifold, and let $a3_1$ and $a3_2$ be strongly irreducible Heegaard surfaces in $M$ that are not isotopic. A lowest genus stabilization is a common stab…

Let $M^n$ be a closed $n$-dimensional manifold with $n\geq 2$, and let $\omega_p^1(M^n,g)$ denote its $1$-dimensional $p$-width with respect to a Riemannian metric $g$. The volume…

Let $M^{n+1}$ be a closed smooth manifold, and let $g$ be a Baire $C^\infty$-generic Riemannian metric on $M$. A min-max $g$-minimal hypersurface is a minimal hypersurface generate…

Let $\Omega\subset\mathbb{R}^n$ be a connected domain with piecewise smooth boundary having $\theta$-corners, where $\theta\in(0,\pi)$. Fix $\eta>0$. For all $p\geq p_0(\Omega)$,…

Let $n>k\geq1$, let $F:X^p\rightarrow\mathcal{Z}_k([0,1]^n,\partial[0,1]^n;\mathbb{Z}_2)$ be a continuous family, and let $r\in(0,\tfrac12)$. There should exist a family…

Let $F:X\rightarrow\mathcal{Z}_k([0,1]^n,\partial[0,1]^n;\mathbb{Z}_2)$ be a contractible family, and let $I_{k+1}([0,1]^n;\mathbb{Z}_2)$ denote the $(k+1)$-chains with mod-2 coeff…

Let $M$ be a closed $3$-manifold, and let $S$ be an embedded surface in $M$ with nontrivial homology. Let $\widehat{C}_{[S]}$ be the lower bound defined using the width of the homo…

Let $\beta_0$ be the min-max width associated with a nontrivial loop in the space of immersions of the round sphere $S^2$ into $\mathbb{R}^3$, equivalently the infimum of the maxim…

Let $M^{n+1}$ be a closed manifold with a generic Riemannian metric, where $3\le n+1\le 7$. Consider closed minimal hypersurfaces obtained by min-max methods, and call a component…

Let $\Omega$ be a compact domain in $\mathbb{R}^{n+1}$ with analytic boundary $\partial\Omega$. A free boundary min-max minimal hypersurface is a free boundary minimal hypersurface…

Let $n$ be the dimension parameter and consider compact smooth domains in $\mathbb{R}^{n+1}$. A free boundary min-max minimal hypersurface is a free boundary minimal hypersurface p…

Let $M^3$ be a Riemannian $3$-manifold, and let a min-max limit arising from a Heegaard splitting have components $\Gamma_i$. Call a component unstable if it is unstable as a minim…

Let $M$ be a Riemannian $3$-manifold, let $H$ be a Heegaard splitting, and let a min-max sequence converge to a limiting minimal surface, possibly with multiplicity. A disk surgery…

Let $X$ be a compact Riemannian manifold of dimension $n$ isometrically embedded in $\mathbb{R}^N$, and let $1\leq k\leq n$. For $\varepsilon>0$, let $[\Pi]$ be a homotopy class of…