微分几何

We construct a form of the $D_4^-$-singularity of fronts in $\R^3$ which uses coordinate transformation on the source and isometry on the target. As an application, we compute differential geometric invariants near the $D_4^-$-singularity,…

Endo-Pajitnov manifolds are compact non-K\"ahler manifolds which generalize the Inoue surfaces $S_M$ to higher dimensions. We compute their Dolbeault cohomology and show that they satisfy the Hodge decomposition at the level of dimensions.

We consider the Tanaka-Webster geometry of surfaces embedded in a 3-dimensional Lie group with a CR structure inherited by a contact form. We define the notions of Gauss and mean curvature and give specific examples.

In this paper, we prove an approximation and interpolation theorem for maxfaces in the Lorentz--Minkowski $3$-space $\mathbb{L}^3$. Alarc\'on, Forstneri\v{c}, and L\'opez established approximation and interpolation theorems for conformal…

The celebrated Chern conjecture asserts that any closed minimal hypersurface in $\mathbb{S}^{n+1}$ with constant scalar curvature is isoparametric. In this paper, we resolve this conjecture in the affirmative for $M^4 \subset \mathbb S^5$…

We give explicit counterexamples to two questions. One is asked by Pogorelov and the other is by Toponogov. These questions concern the existence of closed asymptotic curves in a saddle surface, namely a complete immersed regular surface in…

Gromov's simplicial volume is a fundamental invariant measuring the topological complexity of a manifold. A conjecture of Gromov predicts that every closed manifold admitting a metric of positive scalar curvature has vanishing simplicial…

We study a parabolic obstacle problem for surfaces evolving by anisotropic mean curvature flow subject to an obstacle constraint. Given a convex obstacle and initial data, we seek an evolving surface minimizing an anisotropic energy…

We study an obstacle problem for surfaces minimizing an anisotropic surface energy of ellipsoidal type. Given a convex obstacle and a boundary datum, we seek a surface that minimizes the anisotropic area functional while remaining above the…

We prove a compactness theorem for the space of closed embedded minimal surfaces with area bounded from above and injectivity radius bounded from below in a closed Riemannian $3$-manifold. This result is a variant of the Choi--Schoen…

Let $(M^n,g)$ be a complete Riemannian manifold of dimension $n\geq 5$ endowed with a critical metric of the quadratic scalar-curvature functional $$ \mathcal S^2(g)=\int_M R_g^2\,dV_g . $$ For $n\geq 10$, Catino, Mastrolia and Monticelli…

We construct infinite-time singularities with vanishing mean curvature for Lagrangian mean curvature flow in Gibbons--Hawking spaces. We consider circle-invariant Lagrangian $2$-spheres whose quotient curves are concave and are $C^2$-close…

We study the mean first capture time of isotropic L\'evy flights on Zoll surfaces, namely the expected time for a geodesic L\'evy process to reach a shrinking geodesic ball. While the leading-order asymptotics are universal, we prove that…

The gravitational field of a distant, isolated system is manifested by the conformally invariant Weyl tensor. Thus the conformal structure far from the system encodes the system's gravitational mass. It also encodes the causal structure,…

In this paper we study the structure of complex nilmanifolds $X$ admitting some special classes of Hermitian metrics, namely, astheno-K\"ahler, strongly Gauduchon and balanced metrics. We prove that, in complex dimension 4, the existence of…

We classify, up to conjugacy, the 3-dimensional subalgebras of the Lie algebras associated with the 4-dimensional Thurston geometries whose isometry groups have dimension 4. Since homogeneous hypersurfaces arise as orbits of subgroups of…

We classify polar homogeneous foliations on rank one symmetric spaces of noncompact type up to orbit equivalence.

Let $M^n_\kappa$ be the simply connected space form of dimension $n\ge2$ and constant sectional curvature $\kappa\in\{-1,1\}$. For every bounded connected smooth domain $\Omega\subset M^n_\kappa$, assume in the case $\kappa=1$ that $\Omega$…

A locally conformally K\"ahler (LCK) manifold is a manifold $M$ which admits a K\"ahler structure on its universal cover $\tilde M$, in such a way that the monodromy acts conformally on $\tilde M$. Let $M$ be an $n$-dimensional compact LCK…

For any complete Riemannian manifold $M^n$ with nonnegative Ricci curvature and sublinear diameter growth, we establish a dimensional constraint $n\ge 4s(s-1)+k+1$ if the fundamental group $\pi_1(M)$ contains a torsion-free nilpotent…