37 problems

Let $\Psi_{2,\alpha}$ denote the one-parameter family of degree-2 solutions of the complex sine-Gordon 2 equation constructed in the paper with the $(1,3)$-configuration. Morse-ind…

Let $(N^{n+1},g)$ be a closed Riemannian manifold with positive Ricci curvature. A closed embedded minimal hypersurface is a smooth hypersurface $M^n\subset N^{n+1}$ with zero mean…

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 $\Sigma_a\subset B^3(r(a))\subset\mathbb{H}^3$ be the critical spherical catenoid of the Mori family, a free boundary minimal surface in the geodesic ball, with $a>1/2$. Medved…

Let $\Sigma_a$ be the critical spherical catenoid with parameter $a>1/2$. Decompose its Robin Morse index into Fourier modes $|k|=0$, $|k|=1$, and $|k|\geq 2$. Mode-by-mode index c…

Let $\gamma\in\mathbb{N}$, and let $\Sigma\subset\mathbb{S}^3$ be a closed embedded minimal surface of genus $\gamma$. Denote its Morse index by $\operatorname{Ind}(\Sigma)$, and l…

Let $\vec{\Phi}$ be a Willmore immersion obtained by inverting a minimal immersion with $n$ ends, and suppose that $\vec{\Phi}$ has no branch point. Let…

Let $\vec{\Phi}$ be a Willmore immersion, possibly branched, obtained as the inversion of a minimal immersion with $n$ ends, and let $\operatorname{Ind}_W(\vec{\Phi})$ denote its M…

Index-bounded five-tori conjecture. There exist at least $5$ embedded minimal tori in $(S^3,g)$, each of Morse index at most $9$.

Let $x:M^n\rightarrow\mathbb{S}^{n+1}$ be a full, closed, minimal hypersurface in $\mathbb{S}^{n+1}$, with $n\geq 3$. Here $\operatorname{Ind}(M)$ denotes the Morse index of $M$. P…

Let $M$ be a closed minimal surface in $\mathbb{S}^3$ of genus $\gamma$, and let $\operatorname{ind}(M)$ denote its Morse index. Let $\xi_{\gamma,1}$ be the Lawson surface, whose i…

Clifford hypersurface index conjecture. Equality $\mathrm{Ind}(M)=n+2$ holds precisely for the Clifford hypersurfaces above.

In the energy-supercritical case $p=4$ and dimension $d\geq 5$, consider the singular radial solution $u_{\infty}$ of … for some $\omega_{\infty}\in(d-4,d)$ satisfying … Define the…

Let $b\geq3$ be an integer. A free boundary minimal surface in $\mathbb{B}^3$ is a minimal surface whose boundary lies on and meets $\partial\mathbb{B}^3$ orthogonally. Maximal-are…

Let $n\geq2$ be an integer. A free boundary minimal surface in $\mathbb{B}^3$ is a minimal surface whose boundary lies on and meets $\partial\mathbb{B}^3$ orthogonally. Let…

Let $g\geq2$ be an integer. A free boundary minimal surface in $\mathbb{B}^3$ is a minimal surface whose boundary lies on and meets $\partial\mathbb{B}^3$ orthogonally. Let…

Let $g\geq2$ be an integer. A free boundary minimal surface in $\mathbb{B}^3$ is a minimal surface whose boundary lies on and meets $\partial\mathbb{B}^3$ orthogonally. Let…

Let $\mathbb B^4$ be the $4$-dimensional Euclidean ball, and let the critical Möbius band be the embedded non-orientable free boundary minimal surface in $\mathbb B^4$ whose index…

Let $(M,g)$ be an ambient manifold with positive Ricci curvature, and let $\Sigma$ be an embedded orientable compact minimal hypersurface of index $I$. Marques–Neves' Betti-number…

Let a solution on the hexagonal domain have bend-like vertices, meaning vertices where an associated defect splits from the vertex into an interior defect and a defect at the verte…

Marques–Neves Morse index conjecture. There exists such a hypersurface $\Sigma$ with

Let $(M^n,g)$ be a closed manifold with a generic metric, where $3\leq n\leq 7$. A sequence $\{\Sigma_k\}$ of smooth, embedded, two-sided, closed minimal hypersurfaces is required…

Let $u$ be a weakly harmonic map from $S^3$ into $S^2$, in the framework of weakly harmonic maps considered in the paper. For smooth harmonic maps, Morse index at most $4$ forces i…

Let $(M^{n+1},g)$ be a closed manifold with positive Ricci curvature, and let $\Sigma^n\subset M$ be a closed smoothly embedded minimal hypersurface. Write $b^1(\Sigma)$ for its fi…

Let $m\geq 1$ and let $\overset{\scriptscriptstyle{m}}{{u}}_{p}:=\overset{\scriptscriptstyle{m}}{{u}}_{0,p}$ be the radial solution of the Lane–Emden equation with $\alpha=0$ and D…