Papers

Here is the list of my publications and preprints. The files here may include some minor updates and subsequent corrections to the arXiv versions.

1. The anisotropic Weyl law and Grassmannian equidistribution

   (2026) with Antonio De Rosa.

   PDF · arXiv · cvGMT · Interactive cube-to-slab model
   Abstract: The anisotropic Weyl law and Grassmannian equidistribution

   We establish a Weyl law for the codimension-one anisotropic volume spectrum on compact Riemannian manifolds. We then use this law to obtain equidistribution in the full Grassmannian bundle while keeping the ambient metric fixed. More precisely, on every closed connected smooth Riemannian manifold of dimension three through six, we construct smooth, embedded hypersurfaces stationary for anisotropies smoothly converging to area. The cumulative sum of their area measures, when lifted to the Grassmannian bundle, converges to the uniform measure.

2. On Cylindrical singularities of minimal hypersurfaces

   (2026) with Luca Spolaor.

   PDF · arXiv · cvGMT
   Abstract: On Cylindrical singularities of minimal hypersurfaces

   In a neighborhood of the origin in ℝ⁹, we construct a minimal hypersurface with respect to a smooth conformally Euclidean metric, whose singular set consists of the origin and a sequence of isolated singular points converging to it. At each of the isolated singular points the hypersurface has the unique tangent cone C_4,3, the cone over S⁴(√(4/7)) × S³(√(3/7)), whereas at the origin its unique tangent cone is the cone C_3,3 × ℝ. The metric is Euclidean near each isolated singular point and agrees with the Euclidean metric to infinite order at the origin. A key step, of independent interest, is the construction of an entire stationary hypersurface in Euclidean ℝ⁹ with exactly one singular point at the origin, at which its unique tangent cone is C_4,3, and whose unique tangent cone at infinity is C_3,3 × ℝ.

3. Min-Max construction of anisotropic minimal surfaces with genus bound

   (2026) with Antonio De Rosa and Ling Wang.

   PDF · arXiv · cvGMT
   Abstract: Min-Max construction of anisotropic minimal surfaces with genus bound

   We establish an anisotropic analogue of the celebrated theorem of Meeks–Simon–Yau: every minimizing sequence of surfaces within a fixed isotopy class converges to a smooth stable anisotropic minimal surface, with genus lower semicontinuity. This result also strengthens White’s foundational existence theory for anisotropic minimal disks. As an application, we develop an anisotropic Simon–Smith min–max theory. In every closed 3-manifold, we construct anisotropic min–max sequences within fixed isotopy classes whose limits are stable anisotropic minimal surfaces that are smooth except possibly at a single point. If the integrand satisfies either an ellipticity bound or a C³-pinching condition, we remove the singular point by proving two independent removable singularity theorems for anisotropic minimal surfaces that are smooth and stable away from finitely many points. These removable singularity results also allow to remove the singularities arising in the anisotropic Almgren–Pitts min–max construction in 3-manifolds of De Philippis–De Rosa and in its multiparameter variants.

4. Carleman estimates for stationary Q-valued maps: a variational approach

   (2025) with Luca Spolaor.

   PDF · arXiv · cvGMT
   Abstract: Carleman estimates for stationary Q-valued maps: a variational approach

   We prove a Carleman-type estimate for Dirichlet-stationary multivalued functions and apply it to give a different proof of the optimal dimension of the singular set of Dir-minimizing multivalued functions, originally due to Almgren and to De Lellis–Spadaro.

5. Decay of excess for the abelian Higgs model

   (2024) with Guido de Philippis and Alessandro Pigati. J. Eur. Math. Soc. (JEMS), Online First, 2026.

   PDF · arXiv · cvGMT · Journal
   Abstract: Decay of excess for the abelian Higgs model

   In this article we prove that entire critical points (u, ∇) of the self-dual U(1)-Yang–Mills–Higgs functional E₁, with normalized energy

   $$\frac{E_1(u,\nabla ;B_R)}{2\pi {\omega }_{n-2}{R}^{n-2}}≔\frac{{\int }_{B_R}[{\left|\nabla u\right|}^2+\frac{{(1-{\left|u\right|}^2)}^2}{4}+{\left|F_{\nabla }\right|}^2]}{2\pi {\omega }_{n-2}{R}^{n-2}}\le 1+\tau \left(n\right)$$

   for all R > 0, have unique blow-down. Moreover, we show that they are two-dimensional in ambient dimension 2 ≤ n ≤ 4, or in any dimension n ≥ 2 assuming that (u, ∇) is a local minimizer, thus establishing a co-dimension-two analogue of Savin’s theorem. The main ingredient is an Allard-type improvement of flatness.

6. Quantitative stability for Yang-Mills-Higgs instantons in two dimensions

   (2023). Arch. Ration. Mech. Anal. 248, 88.

   PDF · arXiv · cvGMT · Journal
   Abstract: Quantitative stability for Yang-Mills-Higgs instantons in two dimensions

   We prove that if an N-vortex pair nearly minimizes the Yang-Mills-Higgs energy, then it is second order close to a minimizer. First we use new weighted inequalities in two dimensions and compactness arguments to show stability for sections with some regularity. Second we define a selection principle using a penalized functional and by elliptic regularity and smooth perturbation of complex polynomials, we generalize the stability to all nearly minimizing pairs. With the same method, we also prove the analogous second order stability for nearly minimizing pairs on nontrivial line bundles over arbitrary compact smooth surfaces.

7. New weighted inequalities on two-manifolds

   (2023). Rend. Sem. Mat. Univ. Padova, Online First, 2026.

   PDF · arXiv · cvGMT · Journal
   Abstract: New weighted inequalities on two-manifolds

   We establish a new class of weighted L² Poincaré and elliptic functional inequalities on smooth two-manifolds with explicit constants, for a family of weights satisfying a differential equation. This family includes, in particular, weights comparable to products of positive powers of the geodesic distance to finitely many points. Our primary motivation is the derivation of estimates associated with a weighted Hodge decomposition for one-forms.

Communication of AI-discovered proofs

These papers present proofs discovered through interactions with AI under the authors’ direction. The authors checked and rewrote the proofs to communicate their ideas and insights to the broader mathematical community.

1. Integral curvature estimates and geodesic limits for stable abelian Higgs fields

   (2026) with Marco Badran and Marco A. M. Guaraco.

   PDF · arXiv · cvGMT
   Abstract: Integral curvature estimates and geodesic limits for stable abelian Higgs fields

   We prove that, on closed three-manifolds, the limit interface of a sequence of stable abelian Higgs critical points with uniformly bounded energy is a finite union of smooth closed immersed geodesics, possibly with multiplicity and self-intersections. This follows from an integral estimate for the diffuse curvature, derived from an integral bound on the discrepancy.