20 problems

Vanishing conjecture. For every positive quaternion-Kähler manifold different from $\mathbb{H}P^n$,

Hille–Tamarkin conjecture. The extra logarithmic factor in this estimate can be removed, or even replaced by a logarithmic factor with a negative power.

Let $d\geq2$, $q\in(\frac{d+1}{2},d)$, $p>\frac{(d-1)q}{d-q}$, and let $V\in L^q(\mathbb{R}^d)$ be radial. If $z_j$ are the eigenvalues of $H=-\Delta+V$, write $\delta(z_j)$ for th…

Let $N$ be the dimension, let $mathbb H^N$ be hyperbolic $N$-space, and let $Omegasubsetmathbb H^N$ be a horoconvex domain. Write $Gamma(Omega)$ for its fundamental gap, namely th…

Consider the isospectral-domain example, with eigenvalues $\lambda(k)$ and corresponding lower bounds $\mathrm{GLB}(k)$ for the Laplace operator. The numerical results show a conve…

Let $T$ be a triangle with maximal interior angle $\omega$, and let $C_{\rm st,2}$ denote the stability constant associated with the hybrid high-order method. For any…

Let $\Omega\subset\mathbb{R}^n$ be a bounded convex domain, let $V$ be a convex potential, and consider the Schrödinger operator $-\Delta+V$ on $\Omega$ with Dirichlet boundary con…

Let $(M^n,g,f)$ be a compact Bakry–Émery manifold, possibly with smooth strictly convex boundary, of diameter $D$, satisfying … for $\kappa\in\mathbb R$. Let $\lambda_{p,f}$ be the…

Let $(M^n,g)$ be a compact Riemannian manifold satisfying $\operatorname{Ric}(M)>K$, and let $v$ be a one-form with $\|v\|_\infty<C$. Suppose that there exists $u\in W^{2,p}(M)$ sa…

Integral curvature extension conjecture. Integral curvature versions of the estimate for the general case and of the optimal lower-bound estimate when $H<0$ should also hold.

Let $n\geq 2$, and let $\epsilon>0$ and $K>0$. For an $n$-dimensional closed Riemannian manifold $(M,g)$, write $\lambda_{n+1}(\bar{\Delta}^{E},M)$ for the $(n+1)$-st eigenvalue of…

Let $\Omega\subset\mathbb R^n$ be a bounded convex domain with diameter $D$. Its Dirichlet Laplacian has first two eigenvalues $\lambda_1<\lambda_2$, and the fundamental gap is…

Van den Berg's conjecture. The inequality can be improved, for large diameter $D$, to

Optimal spectral-gap conjecture. The set

Let $\Omega$ be a domain, let $B_0$ tend to infinity, and suppose that … where $\lambda_1(\Omega,B_0)$ is the lowest eigenvalue of the magnetic Laplacian with constant magnetic fie…

Funano and Shioya's conjecture. For every natural number $k$, there exists a positive constant $C_k$ depending only on $k$ such that

Cwikel–Kiltz–Mǎntoiu–Weidl conjecture. The estimate

Cheng's conjecture. Eigenvalue $\Gamma_j$'s of the clamped plate problem (1.1) satisfy an upper bound with optimal order in $j$.

Rearrangement-based Bargmann conjecture.

Let $M$ be a compact spin Kähler manifold of positive scalar curvature $S$, even complex dimension $m$, and minimum scalar curvature $S_0$. For every eigenvalue $\lambda$ of its Di…