arXiv subjects
Publications and source records attributed to Stefano Nardulli.
We compute Simons type equations for parallel mean curvature submanifolds in complex space forms N^n(c), with constant holomorphic sectional curvature c, and product spaces N^n(c)xR. These formulas are then used to characterize some of these submanifolds.
We present a survey on multiplicity results for the Allen--Cahn equation and systems in the singular perturbation regime, emphasizing their geometric interpretation through $\Gamma$-convergence and isoperimetric theory. In the scalar case, the Allen--Cahn functional converges to perimeter, giving rise to minimal and constant-mean-curvature hypersurfaces, while vectorial Allen--Cahn systems lead to multi-phase isoperimetric clusters. The main methodological tool discussed is the photography method, a variational-topological approach based on localized approximate solutions and barycenter maps, which enables one to encode the topology of the ambient manifold into multiplicity results. We compare problems posed on closed manifolds with those on manifolds with boundary, describing the distinct geometric effects induced by Neumann and Dirichlet boundary conditions. The survey highlights both the effectiveness and the limitations of this framework, particularly in the vectorial case, where the lack of a full classification of isoperimetric clusters creates fundamental analytical challenges.
Given a smooth, complete Riemannian manifold $M$ with bounded Ricci curvature and positive injectivity radius, we derive a sharp Sobolev inequality for the embedding of $W^{1,p}(M)$ into $L^{\frac{np}{n-p}}(M)$, when $1\le p< n$. We will first reduce the inequality to functions having support with small enough volume. In turn, we will show that the inequality for small volumes is implied by a first order uniform asymptotic expansion of the isoperimetric profile for $M$, for small volumes. We will then show that such an expansion follows from a local, uniform Sobolev inequality for functions in $W^{1,1}$, having support with small enough diameter.
We provide a multiplicity result for solutions of time-independent Gross-Pitaevskii equations on closed Riemannian manifolds. Such solutions arise as (possibly non-minimizing) critical points of the Ginzburg-Landau energy having prescribed momentum according to a given tangent velocity field. Lower bounds on the multiplicity of solutions are obtained in terms of the topology of the maximum velocity set, in the small momentum and vorticity core size regime. The proof relies on methods from critical point theory and $\Gamma$-convergence for Ginzburg-Landau functionals as well as on some new results for codimension 2 isoperimetric-type problems in the small flux regime, possibly of independent interest.
Given an area-minimizing integral $m$-current in $Σ$, we prove that the Hausdorff dimension of the interior singular set of $T$ cannot exceed $m-2$, provided that $Σ$ is an embedded $(m+\bar{n})$-submanifold of $\mathbb{R}^{m+n}$ of class $C^{2,α}$, where $α>0$. This result establishes the complete counterpart, in the arbitrary codimension setting, of the interior regularity theory for area-minimizing integral hypercurrents within a Riemannian manifold of class $C^{2,α}$.
We prove that in a compact Riemannian manifold, the $m$-minimal clusters of sufficiently small total volume are connected and with small diameter, while in a more general Finsler manifold they are done by at most $m$ connected components of small diameter. We apply these results to calculate the asymptotic expansion of the multi-isoperimetric profile at the first nontrivial order, for small volumes.
In this paper, we prove Allard's Interior $\varepsilon$-Regularity Theorem for $m$-dimensional varifolds with generalized mean curvature in $L^p_{loc}$, $p > m$, in non-collapsed Alexandrov spaces with curvature bounded both from above and below. We first develop an intrinsic proof of the theorem for varifolds in Riemannian manifolds with metric tensor of class $\mathcal{C}^2$, without appealing to Nash's Isometric Embedding Theorem. This yields explicitly computable constants depending only on $m$, $n$, the double sided sectional curvature bounds, and the harmonic radius (or, equivalently, the injectivity radius). We then extend the result to Alexandrov spaces via the Approximation Theorem of Berestovskij and Nikolaev, where the explicit control of the constants in terms of the geometric data is required for the approximation argument.
We extend previous works on the multiplicity of solutions to the Allen-Cahn system on closed Riemannian manifolds by considering an arbitrary number of phases. Specifically, we show that on parallelizable manifolds, the number of solutions is bounded from below by topological invariants of the underlying manifold, provided the temperature parameter and volume constraint are sufficiently small. The Allen-Cahn system naturally arises in phase separation models, where solutions represent the distribution of distinct phases in a multi-component mixture. As the temperature parameter approaches zero, the system's energy approximates the multi-isoperimetric profile, leading to solutions concentrating in regions resembling isoperimetric clusters. For two or three phases, these results rely on classifying isoperimetric clusters. However, this classification remains incomplete for a larger number of phases. To address this technical issue, we employ a "volume-fixing variations" approach, enabling us to establish our results for any number of phases and small volume constraints. This offers deeper insights into phase separation phenomena on manifolds with arbitrary geometry.
In the present work, we consider area minimizing currents in the general setting of arbitrary codimension and arbitrary boundary multiplicity. We study the boundary regularity of 2d area minimizing currents, beyond that, several results are stated in the more general context of $(C_0, α_0, r_0)$-almost area minimizing currents of arbitrary dimension $m$ and arbitrary codimension taking the boundary with arbitrary multiplicity. Furthermore, we do not consider any type of convex barrier assumption on the boundary, in our main regularity result which states that the regular set, which includes one-sided and two-sided points, of any 2d area minimizing current $T$ is an open dense set in the boundary.
This paper deals with quasi-local isoperimetric versions of the positive mass theorem on $3$-manifolds endowed with continuous complete metrics having nonnegative scalar curvature in a suitable weak sense. As a corollary, we derive existence results for isoperimetric sets in such low regularity setting. Our main tool is a new local version of the weak inverse mean curvature flow enjoying $C^0$-stable quantitative estimates.
We present multiplicity results for mass constrained Allen-Cahn equations on a Riemannian manifold with boundary, considering both Neumann and Dirichlet conditions. These results hold under the assumptions of small mass constraint and small diffusion parameter. We obtain lower bounds on the number of solutions according to the Lusternik--Schnirelmann category of the manifold in case of Dirichlet boundary conditions and of its boundary in the case of Neumann boundary conditions. Under generic non-degeneracy assumptions on the solutions, we obtain stronger results based on Morse inequalities. Our approach combines topological and variational methods with tools from Geometric Measure Theory.
We give a multiplicity result for solutions of the Van der Waals-Cahn-Hilliard two-phase transition equation with volume constraints on a closed Riemannian manifold. Our proof employs some results from the classical Lusternik--Schnirelman and Morse theory, together with a technique, the so-called \emph{photography method}, which allows us to obtain lower bounds on the number of solutions in terms of topological invariants of the underlying manifold. The setup for the photography method employs recent results from Riemannian isoperimetry for small volumes.
We prove the existence of multiple solutions to the Allen--Cahn--Hilliard (ACH) vectorial equation (with two equations) involving a triple-well (triphasic) potential with a small volume constraint on a closed parallelizable Riemannian manifold. More precisely, we find a lower bound for the number of solutions depending on some topological invariants of the underlying manifold. The phase transition potential is considered to have a finite set of global minima, where it also vanishes, and a subcritical growth at infinity. Our strategy is to employ the Lusternik--Schnirelmann and infinite-dimensional Morse theories for the vectorial energy functional. To this end, we exploit that the associated ACH energy $Γ$-converges to the weighted multi-perimeter for clusters, which combined with some deep theorems from isoperimetric theory yields the suitable setup to apply the photography method. Along the way, the lack of a closed analytic expression for the multi-isoperimetric function for clusters imposes a delicate issue. Furthermore, using a transversality theorem, we also show the genericity of the set of metrics for which solutions to the ACH system are nondegenerate.
We establish a structure theorem for minimizing sequences for the isoperimetric problem on noncompact $\mathsf{RCD}(K,N)$ spaces $(X,\mathsf{d},\mathcal{H}^N)$. Under the sole (necessary) assumption that the measure of unit balls is uniformly bounded away from zero, we prove that the limit of such a sequence is identified by a finite collection of isoperimetric regions possibly contained in pointed Gromov--Hausdorff limits of the ambient space $X$ along diverging sequences of points. The number of such regions is bounded linearly in terms of the measure of the minimizing sequence. The result follows from a new generalized compactness theorem, which identifies the limit of a sequence of sets $E_i\subset X_i$ with uniformly bounded measure and perimeter, where $(X_i,\mathsf{d}_i,\mathcal{H}^N)$ is an arbitrary sequence of $\mathsf{RCD}(K,N)$ spaces. An abstract criterion for a minimizing sequence to converge without losing mass at infinity to an isoperimetric set is also discussed. The latter criterion is new also for smooth Riemannian spaces.
In this paper we show that, if $T$ is an area-minimizing $2$-dimensional integral current with $\partial T = Q [\![ Γ]\!]$, where $Γ$ is a $C^{1,α}$ curve for $α>0$ and $Q$ an arbitrary integer, then $T$ has a unique tangent cone at every boundary point, with a polynomial convergence rate. The proof is a simple reduction to the case $Q=1$, studied by Hirsch and Marini.
We consider integral area-minimizing $2$-dimensional currents $T$ in $U\subset \mathbb R^{2+n}$ with $\partial T = Q[\![Γ]\!]$, where $Q\in \mathbb N \setminus \{0\}$ and $Γ$ is sufficiently smooth. We prove that, if $q\in Γ$ is a point where the density of $T$ is strictly below $\frac{Q+1}{2}$, then the current is regular at $q$. The regularity is understood in the following sense: there is a neighborhood of $q$ in which $T$ consists of a finite number of regular minimal submanifolds meeting transversally at $Γ$ (and counted with the appropriate integer multiplicity). In view of well-known examples, our result is optimal, and it is the first nontrivial generalization of a classical theorem of Allard for $Q=1$. As a corollary, if $Ω\subset \mathbb R^{2+n}$ is a bounded uniformly convex set and $Γ\subset \partial Ω$ a smooth $1$-dimensional closed submanifold, then any area-minimizing current $T$ with $\partial T = Q [\![Γ]\!]$ is regular in a neighborhood of $Γ$.
We contribute to an original problem studied by Hamilton and others, in order to understand the behaviour of maximal solutions of the Ricci flow both in compact and non-compact complete orientable Riemannian manifolds of finite volume. The case of dimension two has peculiarities, which force us to use different ideas from the corresponding higher dimensional case. We show the existence of connected regions with a connected complementary set (the so-called "separating regions"). In dimension higher than two, the associated problem of minimization is reduced to an auxiliary problem for the isoperimetric profile. This is possible via an argument of compactness in geometric measure theory. Indeed we develop a definitive theory, which allows us to circumvent the shortening curve flow approach of previous authors at the cost of some applications of geometric measure theory and Ascoli-Arzela's Theorem.
We give multiplicity results for the solutions of a nonlinear elliptic equation, with an asymmetric double well potential of Van der Waals-Allen--Cahn--Hilliard type, satisfying a linear volume constraint, on a bounded Lipschitz domain $Ω\subset\mathds R^N$. The number of solutions is estimated in terms of topological and homological invariants of the underlying domain $Ω$.