We consider the flow of closed convex hypersurfaces in Euclidean space $\mathbb{R}^{n+1}$ with speed given by a power of the $k$‑th mean curvature $E_k$ plus a global term chosen to impose a constraint involving the enclosed volume $V_{n+1}$ and the mixed volume $V_{n+1-k}$ of the evolving hypersurface. We prove that if the initial hypersurface is strictly convex, then the solution of the flow exists for all time and converges to a round sphere smoothly. No curvature pinching assumption is required on the initial hypersurface.

For a complex simple simply connected Lie group $G$, and a compact Riemann surface $C$, we consider two sorts of families of flat $G$-connections over $C$. Each family is determined by a point $\mathbf{u}$ of the base of Hitchin’s integrable system for $(G,C)$. One family $\nabla_{\hbar ,\mathbf{u}}$ consists of $G$-opers, and depends on $\hbar \in \mathbb{C}^\times$. The other family $\nabla_{R, \zeta,\mathbf{u}}$ is built from solutions of Hitchin’s equations, and depends on $\zeta \in \mathbb{C}^\times , R \in \mathbb{R}^+$. We show that in the scaling limit $R \to 0, \zeta = \hbar R$, we have $\nabla_{R,\zeta,\mathbf{u}} \to \nabla_{\hbar,\mathbf{u}}$. This establishes and generalizes a conjecture formulated by Gaiotto.

We give a new construction of compact Riemannian $7$-manifolds with holonomy $G_2$. Let $M$ be a torsion-free $G_2$-manifold (which can have holonomy a proper subgroup of $G_2$) such that $M$ admits an involution $\iota$ preserving the $G_2$-structure. Then $M / {\langle \iota \rangle}$ is a $G_2$- orbifold, with singular set $L$ an associative submanifold of $M$, where the singularities are locally of the form $\mathbb{R}^3 \times (\mathbb{R}^4 / {\lbrace \pm 1 \rbrace})$. We resolve this orbifold by gluing in a family of Eguchi–Hanson spaces, parametrized by a nonvanishing closed and coclosed 1-form $\lambda$ on $L$.

Much of the analytic difficulty lies in constructing appropriate closed $G_2$-structures with sufficiently small torsion to be able to apply the general existence theorem of the first author. In particular, the construction involves solving a family of elliptic equations on the noncompact Eguchi–Hanson space, parametrized by the singular set $L$. We also present two generalizations of the main theorem, and we discuss several methods of producing examples from this construction.

We prove that the number of nodal points on an $\mathcal{S}$-good real analytic curve $\mathcal{C}$ of a sequence $\mathcal{S}$ of Laplace eigenfunctions $\varphi_j$ of eigenvalue $-\lambda^2_j$ of a real analytic Riemannian manifold $(M, g)$ is bounded above by $A_{g , \mathcal{C}} \lambda_j$. Moreover, we prove that the codimension-two Hausdorff measure $\mathcal{H}^{m-2} (\mathcal{N}_{\varphi \lambda} \cap H)$ of nodal intersections with a connected, irreducible real analytic hypersurface $H \subset M$ is $\leq A_{g, H} \lambda_j$. The $\mathcal{S}$-goodness condition is that the sequence of normalized logarithms $\frac{1}{\lambda_j} \operatorname{log} {\lvert \varphi_j \rvert}^2$ does not tend to $-\infty$ uniformly on $\mathcal{C}$, resp. $H$. We further show that a hypersurface satisfying a geometric control condition is $\mathcal{S}$-good for a density one subsequence of eigenfunctions. This gives a partial answer to a question of Bourgain–Rudnick about hypersurfaces on which a sequence of eigenfunctions can vanish. The partial answer characterizes hypersurfaces on which a positive density sequence can vanish or just have $L^2$ norms tending to zero.