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Antonio De Rosa

Publications and source records attributed to Antonio De Rosa.

At least 19 recordsLinked to original sources
arXiv2026

Regularity of varifolds with bounded anisotropic first variation

We prove an $\varepsilon$-regularity theorem for $m$-varifolds with mean curvature in $L^p$, $p>m$, with respect to an anisotropic integrand satisfying a Michael-Simon inequality and the quadratic exposed condition: near sufficiently flat density-one points, such varifolds are representable as $C^{1,α}$ graphs. Combined with the recent proof of the anisotropic Michael-Simon inequality, this establishes an anisotropic Allard regularity theorem in arbitrary codimension for a large class of anisotropic integrands, including those close to the area functional. We also exhibit the first examples of anisotropies satisfying both the uniform scalar atomic condition and the Michael-Simon inequality, that are not close to any ellipsoidal norm. These include the $\ell^q$ norms in every dimension and codimension, for explicit ranges of $q$, and a new class of axisymmetric anisotropies.

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arXiv2026

Non-polyconvex $Q$-integrands with lower semicontinuous energies

For any fixed $Q \in \mathbb N$, we construct a positive measure on the space of positively oriented $2$-vectors in $\mathbb R^4$, whose barycenter is a simple $2$-vector, yet which cannot be approximated by weighted Gaussian images of Lipschitz $Q$-graphs. The construction extends to positively oriented $m$-vectors in $\mathbb R^n$ whenever $n- 2 \ge m \ge 2$. This geometric obstruction implies that the approximation result established in [Arch. Ration. Mech. Anal., 2025] is sharp: all $Q \in \mathbb N$ are indeed necessary to ensure the density of weighted Gaussian images of Lipschitz multigraphs in the space of positive measures with simple barycenter. As an application, we prove that for every $Q \ge 1$ and $ p \ge 2$ there exists a non-polyconvex $Q$-integrand whose associated energy is weakly lower semicontinuous in $W^{1,p}$. This also provides new insight into the question posed in [Arch. Ration. Mech. Anal., 2025, Remark 1.14].

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arXiv2026

Lectures on Existence and Regularity of Anisotropic Minimal Surfaces

For several natural phenomena, the use of the surface area functional is a first approximation. In order to capture microstructures, numerous models in applied sciences employ directionally dependent functionals, known as anisotropic energies. Since anisotropic energies are not invariant under rigid motions, their critical points do not enjoy the same conservation laws as isotropic minimal surfaces. For instance, the monotonicity formula for the density ratio is not known to hold for minimizers of general anisotropic energies. Consequently, the study of anisotropic minimal surfaces is more challenging than the study of their isotropic counterparts. In this mini-course, we give an overview of the state of the art in the existence and regularity theory of anisotropic minimal surfaces. In particular, we first focus on solutions of the anisotropic Plateau problem and subsequently move to the investigation of the rectifiability and regularity theory for critical points of anisotropic energies. To conclude, we provide applications to the min-max theory for the construction of closed optimally regular anisotropic minimal hypersurfaces in closed Riemannian manifolds.

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arXiv2026

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 $C^3$-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.

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arXiv2026

On the power of linear programming for K-means clustering

In [SIAM J. Optim., 2022], the authors introduced a new linear programming (LP) relaxation for K-means clustering. In this paper, we further investigate both theoretical and computational properties of this relaxation. As evident from our numerical experiments with both synthetic real-world data sets, the proposed LP relaxation is almost always tight; i.e. its optimal solution is feasible for the original nonconvex problem. To better understand this unexpected behaviour, on the theoretical side, we focus on K-means clustering with two clusters, and we obtain sufficient conditions under which the LP relaxation is tight. We further analyze the sufficient conditions when the input is generated according to a popular stochastic model and obtain recovery guarantees for the LP relaxation. We conclude our theoretical study by constructing a family of inputs for which the LP relaxation is never tight. Denoting by $n$ the number of data points to be clustered, the LP relaxation contains $Ω(n^3)$ inequalities making it impractical for large data sets. To address the scalability issue, by building upon a cutting-plane algorithm together with the GPU implementation of PDLP, a first-order method LP solver, we develop an efficient algorithm that solves the proposed LP and hence the K-means clustering problem, for up to $n \leq 4000$ data points.

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arXiv2025

Anisotropic min-max via phase transitions

We develop a PDE-based approach to the min-max construction of nontrivial integer rectifiable varifolds that are stationary with respect to anisotropic surface energies on closed Riemannian manifolds, in codimension one. Specifically, we study the anisotropic analogue of the Allen-Cahn energy and establish a Modica-type gradient bound for its critical points. Using this in conjunction with certain estimates for stable solutions, we then prove that the energy densities of stable or bounded-Morse-index critical points of its rescalings concentrate along an integer rectifiable varifold that is stationary for the underlying anisotropic integrand. As a consequence, we construct a (possibly singular) anisotropic min-max hypersurface via Allen-Cahn, obtaining an analogue of the result of Hutchinson-Tonegawa in the anisotropic setting.

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arXiv2025

Rigidity of critical points of hydrophobic capillary functionals

We prove the rigidity, among sets of finite perimeter, of volume-preserving critical points of the capillary energy in the half space, in the case where the prescribed interior contact angle is between $90^\circ$ and $120^\circ$. No structural or regularity assumption is required on the finite perimeter sets. Assuming that the ``tangential'' part of the capillary boundary is $\mathcal{H}^n$-null, this rigidity theorem extends to the full hydrophobic regime of interior contact angles between $90^\circ$ and $180^\circ$. Furthermore, we establish the anisotropic counterpart of this theorem under the assumption of lower density bounds.

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arXiv2025

Construction of fillings with prescribed Gaussian image and applications

We construct $d$-dimensional polyhedral chains such that the distribution of tangent planes is close to a prescribed measure on the Grassmannian and the chains are either cycles (if the barycenter of the prescribed measure, considered as a measure on $\bigwedge^d \mathbb{R}^n$, is $0$) or their boundary is the boundary of a unit $d$-cube (if the barycenter of the prescribed measure is a simple $d$-vector). Such fillings were first proved to exist by Burago and Ivanov [Geom. funct. anal., 2004]; our work gives an explicit construction, which is also flexible to generalizations. For instance, in the case that the measure on the Grassmannian is supported on the set of positively oriented $d$-planes, we can construct fillings that are Lipschitz multigraphs. We apply this construction to prove the surprising fact that, for anisotropic integrands, polyconvexity is equivalent to quasiconvexity of the associated $Q$-integrands (that is, ellipticity for Lipschitz multigraphs) and to show that strict polyconvexity is necessary for the atomic condition to hold.

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arXiv2024

Existence and regularity of min-max anisotropic minimal hypersurfaces

In any closed smooth Riemannian manifold of dimension at least three, we use the min-max construction to find anisotropic minimal hyper-surfaces with respect to elliptic integrands, with a singular set of codimension~$2$ vanishing Hausdorff measure. In particular, in a closed $3$-manifold, we obtain a smooth anisotropic minimal surface. The critical step is to obtain a uniform upper bound for density ratios in the anisotropic min-max construction. This confirms a conjecture by Allard [Invent. Math., 1983].

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arXiv2024

The ratio-cut polytope and K-means clustering

We introduce the ratio-cut polytope defined as the convex hull of ratio-cut vectors corresponding to all partitions of $n$ points in $\mathbb R^m$ into at most $K$ clusters. This polytope is closely related to the convex hull of the feasible region of a number of clustering problems such as K-means clustering and spectral clustering. We study the facial structure of the ratio-cut polytope and derive several types of facet-defining inequalities. We then consider the problem of K-means clustering and introduce a novel linear programming (LP) relaxation for it. Subsequently, we focus on the case of two clusters and derive sufficient conditions under which the proposed LP relaxation recovers the underlying clusters exactly. Namely, we consider the stochastic ball model, a popular generative model for K-means clustering, and we show that if the separation distance between cluster centers satisfies $Δ> 1+\sqrt 3$, then the LP relaxation recovers the planted clusters with high probability. This is a major improvement over the only existing recovery guarantee for an LP relaxation of K-means clustering stating that recovery is possible with high probability if and only if $Δ> 4$. Our numerical experiments indicate that the proposed LP relaxation significantly outperforms a popular semidefinite programming relaxation in recovering the planted clusters.

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arXiv2024

Boundary regularity for anisotropic minimal Lipschitz graphs

We prove that $m$-dimensional Lipschitz graphs in any codimension with $C^{1,α}$ boundary and anisotropic mean curvature bounded in $L^p$, $p > m$, are regular at every boundary point with density bounded above by $1/2 +σ$, provided the anisotropic energy satisfies the uniform scalar atomic condition.

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arXiv2023

Explicit convex hull description of bivariate quadratic sets with indicator variables

We consider the nonconvex set $\mathcal S_n = \{(x,X,z): X = x x^T, \; x (1-z) =0,\; x \geq 0,\; z \in \{0,1\}^n\}$, which is closely related to the feasible region of several difficult nonconvex optimization problems such as the best subset selection and constrained portfolio optimization. Utilizing ideas from convex analysis and disjunctive programming, we obtain an explicit description for the closure of the convex hull of $\mathcal S_2$ in the space of original variables. In order to generate valid inequalities corresponding to supporting hyperplanes of the convex hull of $\mathcal S_2$, we present a simple separation algorithm that can be incorporated in branch-and-cut based solvers to enhance the quality of existing relaxations.

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arXiv2022

Efficient Joint Object Matching via Linear Programming

Joint object matching, also known as multi-image matching, namely, the problem of finding consistent partial maps among all pairs of objects within a collection, is a crucial task in many areas of computer vision. This problem subsumes bipartite graph matching and graph partitioning as special cases and is NP-hard, in general. We develop scalable linear programming (LP) relaxations with theoretical performance guarantees for joint object matching. We start by proposing a new characterization of consistent partial maps; this in turn enables us to formulate joint object matching as an integer linear programming (ILP) problem. To construct strong LP relaxations, we study the facial structure of the convex hull of the feasible region of this ILP, which we refer to as the joint matching polytope. We present an exponential family of facet-defining inequalities that can be separated in strongly polynomial time, hence obtaining a partial characterization of the joint matching polytope that is both tight and cheap to compute. To analyze the theoretical performance of the proposed LP relaxations, we focus on permutation group synchronization, an important special case of joint object matching. We show that under the random corruption model for the input maps, a simple LP relaxation, that is, an LP containing only a very small fraction of the proposed facet-defining inequalities, recovers the ground truth with high probability if the corruption level is below $40\%$. Finally, via a preliminary computational study on synthetic data, we show that the proposed LP relaxations outperform a popular SDP relaxation both in terms of recovery and tightness.

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arXiv2022

The anisotropic Min-Max theory: Existence of anisotropic minimal and CMC surfaces

We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth $3$-dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard [Invent. Math.,1983] in dimension $3$.

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arXiv2021

Absence of bubbling phenomena for non convex anisotropic nearly umbilical and quasi Einstein hypersurfaces

We prove that, for every closed (not necessarily convex) hypersurface $Σ$ in $\mathbb{R}^{n+1}$ and every $p>n$, the $L^p$-norm of the trace-free part of the anisotropic second fundamental form controls from above the $W^{2,p}$-closeness of $Σ$ to the Wulff shape. In the isotropic setting, we provide a simpler proof. This result is sharp since in the subcritical regime $p\leq n$, the lack of convexity assumptions may lead in general to bubbling phenomena. Moreover, we obtain a stability theorem for quasi Einstein (not necessarily convex) hypersurfaces and we improve the quantitative estimates in the convex setting.

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arXiv2020

Regularity for graphs with bounded anisotropic mean curvature

We prove that $m$-dimensional Lipschitz graphs with anisotropic mean curvature bounded in $L^p$, $p>m$, are regular almost everywhere in every dimension and codimension. This provides partial or full answers to multiple open questions arising in the literature. The anisotropic energy is required to satisfy a novel ellipticity condition, which holds for instance in a $C^2$ neighborhood of the area functional. This condition is proved to imply the atomic condition. In particular we provide the first non-trivial class of examples of anisotropic energies in high codimension satisfying the atomic condition, addressing an open question in the field. As a byproduct, we deduce the rectifiability of varifolds (resp. of the mass of varifolds) with locally bounded anisotropic first variation for a $C^2$ (resp. $C^1$) neighborhood of the area functional. In addition to these examples, we also provide a class of anisotropic energies in high codimension, far from the area functional, for which the rectifiability of the mass of varifolds with locally bounded anisotropic first variation holds. To conclude, we show that the atomic condition excludes non-trivial Young measures in the case of anisotropic stationary graphs.

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