We study the singular $SU(3)$ Toda system on a torus\[\begin{cases}\Delta u + 2e^u - e^v = 4 \pi n \delta_0 \\\Delta v + 2e^v - e^u = 4 \pi n_2 \delta_0 \\\end{cases}\quad \textrm{on} \quad E_\tau := \mathbb{C}/(\mathbb{Z}+\mathbb{Z} \tau) \: ,\]where $\delta_0$ is the Dirac measure at $0$ and $n, n_2 \in \mathbb{N}$. For the noncritical case $3 \nmid (n_2 - n)$ where solutions always exists by [Battaglia et al. Adv. Math. 2015], we proved recently in [Chen-Lin, J. Differ. Geom. 2023] that the number of solutions is at most $(n + 1)(n_2 + 1)(n + n_2 + 2)/6$. <p>This paper is the first one to study the critical case $n_2 = n+3l$ with $l \in \mathbb{N}$, for which the a priori estimates do not hold and the (non-)existence of solutions has been a longstanding problem. We show that

(i) When $n, l$ are both odd, the Toda system has no even solutions.

(ii) When $n$ is odd and $l$ is even, the Toda system always has even solutions, and each even solution belongs to a unique $2$-parametric family of even solutions $(u_{\lambda,\mu})$, where $\lambda,\mu \gt 0$ can be arbitrary. Moreover, if $\tau \in i \mathbb{R}_{\gt 0}$, i.e. $E_\tau$ is a rectangular torus, then the number of $2$-parametric families of even solutions is exactly $\frac{n+1}{2}$ and all these even solutions are axisymmetric. The asymptotics of $(u_{\lambda,\mu}, v_{\lambda,\mu})$ as $\lambda, \mu \to 0, + \infty$ are also studied.

The case that $n$ is even is quite different and the existence of even solutions depends on the geometry of the torus $E_\tau$. As applications, we obtain sharp existence and non-existence results for the $SU(3)$ Toda system with four singular sources in the plane.

We consider the curve shortening flow applied to a class of figure-eight curves: those with dihedral symmetry, convex lobes, and a monotonicity assumption on the curvature. We prove that when (non-conformal) linear transformations are applied to the solution so as to keep the bounding box the unit square, the renormalized limit converges to a quadrilateral $\bowtie$ which we call a bowtie. Along the way we prove that suitably chosen arcs of our evolving curves, when suitably rescaled, converge to the Grim Reaper Soliton under the flow. Our Grim Reaper Theorem is an analogue of a theorem of S. Angenent in $\href{https://dx.doi.org/10.4310/jdg/1214446558}{[2]}$, which is proven in the locally convex case.

Motivated by work of Thurston on defining a version of Teichmüller theory based on best Lipschitz maps between surfaces, we study infinity-harmonic maps from a hyperbolic manifold to the circle. The best Lipschitz constant is taken on a geodesic lamination. Moreover, in the surface case the dual problem leads to a function of least gradient which defines a transverse measure on the lamination. We also discuss the construction of least gradient functions from transverse measures via primitives to Ruelle–Sullivan currents.

We define a family of symplectic invariants which obstruct exact symplectic embeddings between Liouville manifolds, using the general formalism of linearized contact homology and its $\mathcal{L}_\infty$ structure. As our primary application, we investigate embeddings between normal crossing divisor complements in complex projective space, giving a complete characterization in many cases. Our main embedding results are deduced explicitly from pseudoholomorphic curves, without appealing to Hamiltonian or virtual perturbations.

Uniform $L^1$ and lower bounds are obtained for the Green’s function on compact Kähler manifolds. Unlike in the classic theorem of Cheng–Li for Riemannian manifolds, the lower bounds do not depend directly on the Ricci curvature, but only on integral bounds for the volume form and certain of its derivatives. In particular, a uniform lower bound for the Green’s function on compact Kähler manifolds is obtained which depends only on a lower bound for the scalar curvature and on an $L^q$ norm for the volume form for some $q \gt 1$. The proof relies on auxiliary Monge–Ampère equations, and is fundamentally non-linear. The lower bounds for the Green’s function imply in turn $C^1$ and $C^2$ estimates for complex Monge–Ampère equations with a sharper dependence on the function on the right hand side.

We give examples of hyperbolic groups which contain subgroups that are of type $\mathscr{F}_3$ but not of type $\mathscr{F}_4$. These groups are obtained by Dehn filling starting from a non-uniform lattice in $\mathrm{PO}(8, 1)$ which was previously studied by Italiano, Martelli and Migliorini.

$\def\M{\mathcal{M}}\def\Mtilde{\widetilde{\M}}$We develop a virtual cycle theoretic approach towards generalized Donaldson–Thomas theory of Calabi–Yau threefolds. Let $\M$ be the moduli stack of Gieseker semistable sheaves of fixed topological type on a Calabi–Yau threefold $W$.

We construct an associated Deligne–Mumford stack $\Mtilde$ with an induced semi-perfect obstruction theory of virtual dimension zero and define the generalized Donaldson–Thomas invariant of $W$ via Kirwan blowups to be the degree of the virtual cycle $[\Mtilde]^{\mathrm{vir.}}$ We show that it is invariant under deformations of the complex structure of $W$.

Let $X$ be a compact Kähler manifold and $\alpha$ be a Dolbeault cohomology class of bi-degree $(1, 1)$ on $X$. When the numerical dimension of $\alpha$ is one and $\alpha$ admits at least two smooth semipositive representatives, we show the existence of a family of real analytic Levi-flat hypersurfaces in $X$ and a holomorphic foliation on a suitable domain $W$ of $X$ along whose leaves any semi-positive representative of $\alpha$ is zero. We also investigate the maximal choice of such a domain $W$ by considering the analytic continuation of the holomorphic foliation. As an application, we give a criterion for the $U(1)$-flatness of line bundles which particularly gives an affirmative answer to [K2, Conjecture 2.1] on the relation between the semi-positivity of the line bundle $[Y]$ and the analytic structure of a neighborhood of $Y$ for a smooth connected hypersurface $Y$ of $X$.

We show that closed subsets with vanishing first homology in two-dimensional spaces inherit the upper curvature bound from their ambient spaces and discuss topological applications.

In this work we prove that the Whitehead manifold has no complete metric of non-negative scalar curvature. This result can be generalized to the genus one case. Precisely, we show that no contractible genus one 3-manifold admits a complete metric of non-negaitve scalar curvature.

The notions of discrete conformality for polyhedral surfaces have rich mathematical theories and a wide range of applications. The related notions of discrete uniformizations are naturally important in theory, and also very useful in numerically computing the smooth uniformizations. This paper proves that the discrete uniformizations approximate the continuous uniformizations for closed Riemannian surfaces of genus $\geq 1$, if the approximating triangulations are reasonably regular. To the best of the authors’ knowledge, this is the first convergence result on numerically computing uniformization metrics for surfaces of genus $\gt 1$.