5 problems

Let $B$ be a bounded region in $\mathbb{R}^r$, let $r\geq 3$, and let $(\cdot,\cdot)$ be the positive-definite integral quadratic form used to define the frequency sphere. For coef…

The optimal eigenfunction-bound conjecture. This estimate should hold for every $p>2+\frac{4}{d-2}$, with an $\varepsilon$-loss if $2\leq r\leq 4$. The paper establishes evidence f…

Let $\boldsymbol{\Sigma}$ be a compact interior hypersurface of a Riemannian manifold $(M,g)$. Let $\Delta_\Sigma$ be the Laplace–Beltrami operator induced on $\Sigma$, and define…

Let $(M,g)$ be a Riemannian manifold and let $\boldsymbol{\Sigma}$ be an interior hypersurface with positive definite second fundamental form satisfying the transversal geometric c…

Let $(M,g)$ be a Riemannian manifold and let $\boldsymbol{\Sigma}$ be an interior hypersurface with positive definite second fundamental form. Let $(\lambda,\phi)$ satisfy the eige…