Let $\Omega \subset \mathbb{R}^2$ be a bounded piecewise smooth domain and $\varphi \lambda$ be a Neumann (or Dirichlet) eigenfunction with eigenvalue $\lambda^2$ and nodal set $\mathcal{N}_{\varphi \lambda} = \{ x \in \Omega ; \varphi \lambda (x) = 0 \}$. Let $H \subset \Omega$ be an interior $C^{\omega}$ curve. Consider the intersection number \[ n(\lambda, H) := \# (H \cap \mathcal{N}_{\varphi \lambda}). \] We first prove that for general piecewise-analytic domains, and under an appropriate “goodness” condition on $H$ (see Theorem 1.1), \[ n(\lambda, H) = \mathcal{O}_H (\lambda) \] as $\lambda \to \infty$. Then, using Theorem 1.1, we prove in Theorem 1.2 that the bound in the above equation is satisfied in the case of quantum ergodic (QE) sequences of interior eigenfunctions, provided $\Omega$ is convex and $H$ has strictly positive geodesic curvature.

We show that if two gradient shrinking Ricci solitons are asymptotic along some end of each to the same regular cone $((0,\infty) \times \Sigma, dr^2 + r^2 g_{\Sigma} )$, then the soliton metrics must be isometric on some neighborhoods of infinity of these ends. Our theorem imposes no restrictions on the behavior of the metrics off of the ends in question and in particular does not require their geodesic completeness. As an application, we prove that the only complete connected gradient shrinking Ricci soliton asymptotic to a rotationally symmetric cone is the Gaussian soliton on $\mathbb{R}^n$.

We study the issue of connectedness at infinity of gradient Kähler Ricci solitons. For shrinking Kähler Ricci solitons, we show they must be connected at infinity. We also show the same holds true for expanding Kähler Ricci solitons with proper potential functions. As a separate issue, we obtain a sharp pointwise lower bound for the weight function of any smooth metric measure space, in terms of a lower bound of the associated Bakry-Émery curvature.

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts in F. Almgren, The theory of varifolds, and J. Pitts, Existence and regularity of minimal surfaces on Riemannian manifold, corresponding to the fundamental class of a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature with $2 \leq n \leq 6$. We characterize the Morse index, area, and multiplicity of this min-max hypersurface. In particular, we show that the min-max hypersurface is either orientable and of index one, or is a double cover of a non-orientable minimal hypersurface with least area among all closed embedded minimal hypersurfaces.

We prove a generalized Mariño-Vafa formula for Hodge integrals over $\overline{\mathcal{M}}_{g, \gamma - \mu} (\mathcal{B}G)$ with $G$ an arbitrary finite abelian group. This formula can be viewed as a formula for the one-leg orbifold Gromov-Witten vertex where the leg is effective. We will prove the orbifold Gromov-Witten/Donaldson-Thomas correspondence between our formula and the formula for the orbifold DT vertex in J. Bryan, C. Cadman & B. Young, The orbifold topological vertex. We will also use this formula to study the local Gromov-Witten theory of an orbi-curve with cyclic stack points in a Calabi-Yau three-orbifold.