Colding–Minicozzi conjectures on the frequency of harmonic functions

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 function $u$. A polynomial growth harmonic function is one whose growth is bounded by a polynomial in the distance. Colding–Minicozzi conjectures. (a) If $u$ is a non-constant polynomial growth harmonic function on $M^n$, then $\mathscr{F}_u(r)$ is uniformly bounded. (b) Suppose $B_{2r}(p)\subseteq M^n$ for some $p\in M^n$, and $u$ is harmonic on $B_{2r}(p)$ with

$$\sup_{s\in [r, 2r]}\mathscr{F}_u(s)\leq d,$$

where $d>0$ is a constant. Then there exists $C=C(n,d,\mathrm{V}_M)$ such that

$$\sup_{s\in [0,r]}\mathscr{F}_u(s)\leq C.$$

These conjectures concern uniform frequency bounds and quantitative strong unique continuation for harmonic functions. The source presents them as conjectures of Colding and Minicozzi; no resolution is given here.