56 problems

Let $(M^{n+1},g)$ be a closed Riemannian manifold with $3\leq n+1\leq 7$, and let $\omega_k(M^{n+1},g)$ be its min–max widths. For a generic metric $g$, write a min–max realization…

A closed Riemannian manifold $(M^{n+1},g)$ is a compact manifold without boundary equipped with a Riemannian metric. Yau's conjecture. Every closed manifold should contain infinite…

Let $(M^3,g)$ be a closed three-dimensional Riemannian manifold. Yau's conjecture. $M$ contains infinitely many immersed minimal surfaces. In recent years, the development of min-m…

Let $M$ be a compact $n$-dimensional Riemannian manifold, and let $\omega_p^1$ denote the Almgren–Pitts min-max value for $p$-sweepouts through 1-cycles. Gromov's Weyl law conjectu…

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…

Min-max uniqueness conjecture. Under a certain metric on $\mathbb{RP}^3$, every minimal real projective plane constructed via min-max could be the same area-minimizing one, with po…

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 $M$ be a closed Riemannian manifold. Yau's conjecture. The manifold $M$ contains infinitely many minimal surfaces. This is a foundational existence conjecture in the theory of…

Let $(M^3,g)$ be a closed Riemannian $3$-manifold, and let $\{\Sigma_p\}_{p\in\mathbb{N}}$ be the sequence of closed embedded minimal surfaces constructed by the min-max theory, wi…

Let $(M,g)$ be a generic Riemannian manifold. A minimal surface is two-sided if it admits a globally defined unit normal, and its Morse index is the maximal dimension of a subspace…

Let $(M^{n+1},\partial M)$ be a compact Riemannian manifold with smooth boundary. A free boundary minimal hypersurface (FBMH) is a smooth embedded hypersurface…

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 $(M,g)$ be a closed Riemannian manifold with a bumpy metric or with positive Ricci curvature. For an integer $p\geq 2$, a min-max minimal hypersurface is a minimal hypersurface…

Let $P$ be a polygon in $\mathbb{R}^2$, and let a side mean an edge of $P$. For a positive integer $p$, let $\omega_p(P)$ denote the $p$-width of $P$. The single-side $p$-width con…

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…

Let $M$ be a Riemannian 3-torus, let $\waist_1(M)$ denote the infimum, over continuous maps from $M$ to $\mathbb{R}^2$ with Lipschitz 1-cycle fibers, of the supremum of the lengths…

Let $(M^3,g)$ be a closed Riemannian 3-manifold. Yau's conjecture. The manifold $M$ should contain infinitely many closed minimal surfaces. This conjecture concerns the abundance o…

Let $G$ act on a Riemannian manifold, and let a $G$-invariant Riemannian metric be generic. Consider the minimal $G$-hypersurface constructed from $k$-parameter families of equivar…

Let $(M^{n+1},g)$ be a closed Riemannian manifold, and let $\tilde c(p)$ denote its phase-transition half-volume spectrum. A metric is generic if it belongs to the generic class in…

Let $(M^{n+1},g)$ be a closed Riemannian manifold. Denote by $\tilde \omega_p$ its Almgren–Pitts half-volume spectrum and by $\tilde c(p)$ its phase-transition half-volume spectrum…

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 with a bumpy metric, and let $a3_1$ be a strongly irreducible Heegaard surface in $M$. Let $Flip(M,a3_1)$ denote the genus associated with flipp…