We show that for each fixed dimension $d \geq 2$, the set of $d$-dimensional klt elliptic varieties with numerically trivial canonical bundle is bounded up to isomorphism in codimension one, provided that the torsion index of the canonical class is bounded and the elliptic fibration admits a rational section. This case builds on an analogous boundedness result for the set of rationally connected log Calabi–Yau pairs with bounded torsion index. In dimension $3$, we prove the more general statement that the set of $\epsilon \textrm{-lc}$ pairs $(X,B)$ with $-(K_X + B)$ nef and rationally connected $X$ is bounded up to isomorphism in codimension one.

We prove that Klein quartic maximizes the multiplicity of the first positive eigenvalue of the Laplacian among all closed hyperbolic surfaces of genus $3$, with multiplicity equal to $8$. We also obtain partial results in genus $2$, where we find that the maximum multiplicity is between $3$ and $6$. Along the way, we show that for every $g \geq 2$, there exists some $\delta_g \gt 0$ such that the multiplicity of any eigenvalue of the Laplacian on a closed hyperbolic surface of genus $g$ in the interval $[0, 1/4 + \delta_g]$ is at most $2g - 1$ despite the fact that this interval can contain arbitrarily many eigenvalues. This extends a result of Otal to a larger interval but with a weaker bound, which nevertheless improves upon the general upper bound of Sévennec.

We show that conformal geodesics on a Riemannian manifold cannot spiral: there does not exist a conformal geodesic which becomes trapped in every neighbourhood of a point.

We present a connection between minimal surfaces of index one and General Relativity. First, we show that for a certain class of electrostatic systems, each of its unstable horizons is the solution of a one-parameter min‑max problem for the area functional, in particular it has index one. Combining this with a theorem by Marques and Neves we obtain a uniqueness result for electrostatic systems. We also prove an inequality relating the area and the charge of a minimal surface of index one in a Cauchy data satisfying the Dominant Energy Condition.

We study the asymptotics of Ohsawa–Takegoshi extension operator and orthogonal Bergman projector associated with high tensor powers of a positive line bundle.

More precisely, for a fixed complex submanifold in a complex manifold, we consider the operator which associates to a given holomorphic section of a positive line bundle over the submanifold the holomorphic extension of it to the ambient manifold with the minimal $L^2$-norm. When the tensor power of the line bundle tends to infinity, we obtain an explicit asymptotic expansion of this operator. This is done by proving an exponential estimate for the associated Schwartz kernel and showing that this Schwartz kernel admits a full asymptotic expansion. We prove similar results for the projection onto holomorphic sections orthogonal to those which vanish along the submanifold.

$\def\S{\mathbb{S}} \def\M{\mathcal{M}}$ Given a function $f \gt 0$ on the unit sphere $\S^{n+1}$, the Minkowski problem in the sphere concerns the existence of convex hypersurfaces $\M \subset \S^{n+1}$ such that the Gauss curvature of $\M$ at $z$ is equal to $f(\nu(z))$, where $\nu(z)$ is the unit outer normal of $\M$ at $z$. We use the min‑max principle and the Gauss curvature flow to prove that there are at least two solutions to the problem. By using the rotating plane method in the sphere, we also show the existence of a rotationally symmetric and monotone function $f$ such that there are exactly two solutions.

Using non-Abelian Hodge correspondence for parabolic Higgs bundles and surface group representations, we construct infinitely many non-congruent hyperbolic affine spheres modeled on a thrice-punctured sphere with monodromy in $\mathbf{SL}_3 (\mathbb{Z})$. These give rise to non-isometric semi-flat Calabi–Yau metrics on special Lagrangian torus bundles over an open ball in $\mathbb{R}^3$ with a $\mathrm{Y}$-vertex deleted, thereby answering a question raised by Loftin, Yau, and Zaslow in [$\href{https://dx.doi.org/10.4310/jdg/1143644314}{32}$, $\href{https://doi.org/10.48550/arXiv.math/0405061}{33}$].

We introduce dual curvature measures for $\log$-concave functions, which in the case of characteristic functions recover the dual curvature measures for convex bodies introduced by Huang–Lutwak–Yang–Zhang in 2016. Variational formulas are shown. The associated Minkowski problem for these dual curvature measures is considered and sufficient conditions in the symmetric setting are demonstrated.

We classify the (semi-simple parts of the) Lie algebra of the Zariski closure of a discrete subgroup of a split simple real-algebraic Lie group, whose limit sets are minimal and such that the limit set in the space of full flags contains a positive triple of flags (as in Lusztig [23]). We then apply our result to obtain a new proof of Guichard’s classification [17] of Zariski closures of Hitchin representations into $\mathsf{PSL}_d (\mathbb{R})$.

We prove a recent conjecture of Chi Li relating the notion of higher Lelong numbers to that of full Monge–Ampère mass.