112 problems

Let $u$ be a harmonic function on the unit ball in $\mathbb{R}^{n}$ such that $u(0)=0$, and let $\mathcal{H}^{n-1}$ denote the Hausdorff measure of its nodal set. Nadirashvili's co…

A polynomial growth harmonic function on a Riemannian manifold is a harmonic function whose growth rate is bounded by a polynomial; let $d$ be a nonnegative growth-rate parameter,…

Let $\Omega$ be a domain with boundary $M$, and let $u_\lambda$ be a Steklov eigenfunction with eigenvalue $\lambda$. For a point at interior distance $t$ from the boundary, assume…

Yau's finite-dimensionality conjecture. These spaces are finite dimensional.

Let $\Omega\subset\mathbb{R}^d$ be a Lipschitz domain, let $u$ be a bounded harmonic function in $\Omega$, and let $U\subset\partial\Omega$ be relatively open. Lin's conjecture. If…

Let $u$ be a bounded harmonic function on the unit ball … For each $n$, consider the sharp constants in the corresponding estimates for the radial derivative and for the full gradi…

A probability measure on a group is non-degenerate if its support is not contained in a proper subgroup. A probability measure is Liouville when every bounded harmonic function for…

Let $(M^n,g)$, with $n\geq 3$, be a compact Riemannian manifold with nonnegative Ricci curvature, and let the second fundamental form satisfy $\Pi\geq 1$ on $\partial M$. Let…

Logunov–Malinnikova's conjecture. Since the critical set where $| abla f|=0$ has codimension $2$, the correct condition on $E$ should be that it has positive $(n-2+delta)$-dimensio…

Lyons–Sullivan conjecture. The group $\Gamma$ is of exponential growth if and only if $M$ admits non-constant, positive harmonic functions.

Let $\Omega\subset\mathbb{C}$ be a bounded domain whose boundary consists of finitely many non-intersecting Jordan curves. For each polynomial boundary datum on $\partial\Omega$, c…

Let $M^n$ be a complete Riemannian manifold with nonnegative Ricci curvature and maximal volume growth, and let $\mathscr{F}_u(r)$ denote the frequency function of a harmonic funct…

Greene–Wu conjecture. There exists an entire bounded nonconstant harmonic function on $M$. This conjecture concerns the existence of nonconstant bounded harmonic functions on Carta…

Let $h^{\infty}$ denote the space of bounded harmonic functions on the unit ball $\mathbb{B}^n:=\{x\in\mathbb{R}^n:|x|<1\}$. For $x\in\mathbb{B}^n$ and $l\in\mathbb{S}^{n-1}$, let…

Let $M$ be a complete Riemannian manifold with nonnegative Ricci curvature, and fix a growth rate $d$. Consider the space of harmonic functions on $M$ having polynomial growth of r…

Let $M$ be a complete Riemannian manifold with bounded geometry, and suppose its first eigenvalue is positive. A bounded harmonic function on $M$ is called nontrivial when it is no…

Iversen-type conjecture. If the critical set of $f$ is nowhere dense and the cluster set of $f$ has empty interior, then every cluster point of $f$ is an asymptotic value. In parti…

Yor's classification conjecture. The only $(N,\overline{N})$-harmonic functions are

Converse Liouville property conjecture. Any amenable measured groupoid is Liouville.

Let $S$ denote the critical set associated with $f$, and let … where $p(z)$ is a polynomial in $z$ of degree at least two. Let $\{R_j\}$ be a finite collection of bounded regions i…

Let $M$ be a properly embedded minimal surface in $\mathbb{R}^3$. Recall that a Riemannian surface is recurrent if almost all Brownian paths are dense in $M$. Meeks's recurrence co…

Let $\mathcal{I}_e$ denote the set of extremal invariant measures, and let $\hat{\mathcal{H}}_R$ be the relevant class of functions associated with the set $\hat{\mathcal{H}}$ of f…

A group is Choquet–Deny if every bounded harmonic function is constant for every probability measure on it. A group has exponential growth if the size of balls in a finitely genera…

Yau's conjecture. The space $\mathcal{H}_d(M)$ should have finite dimension.

Let $B\subset\mathbb{R}^{n}$ be a unit ball with $n\geq 3$. Let $u:4B\to\mathbb{R}$ be harmonic and satisfy $u(0)=0$, and let $N_u(0,\frac{1}{2})$ denote its doubling index at the…