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Costante Bellettini

Publications and source records attributed to Costante Bellettini.

At least 19 recordsLinked to original sources
arXiv2026

Regularity and compactness theory of stable free-boundary minimal hypersurfaces

We establish the free-boundary version of Schoen-Simon's regularity and compactness theory for stable minimal hypersurfaces in a complete $(n+1)$-dimensional Riemannian manifold-with-boundary, following the intrinsic PDE scheme introduced by the first-named author. The key step is to prove sheeting theorems in the two possible scenarios in which local weak flatness can occur near a point on $\partial M$: flatness with respect to $\partial M$, and flatness with respect to a reference hypersurface-with-boundary meeting $\partial M$ orthogonally. We obtain these theorems via $\varepsilon$-regularity results for geometrically adapted tilt functions that quantify flatness while keeping track of the ambient geometry. These tilt functions satisfy nonlinear elliptic PDEs on the hypersurface which are amenable to an intrinsic analysis via De Giorgi iteration, made possible by the weak Caccioppoli-type inequalities obtained from stability. Both results apply to immersed hypersurfaces with a singular set of locally finite $(n-2)$-dimensional Hausdorff measure. When specialised to embeddings, they yield compactness and regularity: a sequence of stable free-boundary embedded minimal hypersurfaces with uniformly bounded area and singular sets of locally finite $(n-2)$-measure admits a subsequence converging to a free-boundary minimal hypersurface that is smoothly embedded away from a singular set of Hausdorff dimension at most $n-7$. Convergence is smooth and graphical, possibly with multiplicity, away from this singular set. As an application, we extend to arbitrary dimension the free-boundary Almgren-Pitts existence theory developed by the second-named author with X. Zhou and prove that every compact Riemannian manifold-with-boundary of dimension $n+1$ contains a free-boundary minimal hypersurface smoothly embedded outside a singular set of Hausdorff dimension at most $n-7$.

math.DG↗
arXiv2026

On isolated singularities and generic regularity of min-max CMC hypersurfaces

In compact Riemannian manifolds of dimension 3 or higher with positive Ricci curvature, we prove that every constant mean curvature hypersurface produced by the Allen-Cahn min-max procedure of Bellettini-Wickramasekera (with constant prescribing function) is a local minimiser of the natural area-type functional around each isolated singularity. In particular, every tangent cone at each isolated singularity of the resulting hypersurface is area-minimising. As a consequence, for any real $λ$ we show, through a surgery procedure, that for a generic 8-dimensional compact Riemannian manifold with positive Ricci curvature there exists a closed embedded smooth hypersurface of constant mean curvature $λ$; the minimal case ($λ$ = 0) of this result was obtained in work by Chodosh-Liokumovich-Spolaor.

math.DG↗
arXiv2025

Smooth approximations for constant-mean-curvature hypersurfaces with isolated singularities

We consider a CMC hypersurface with an isolated singular point at which the tangent cone is regular, and such that, in a neighbourhood of said point, the hypersurface is the boundary of a Caccioppoli set that minimises the standard prescribed-mean-curvature functional. We prove that in a ball centred at the singularity there exists a sequence of smooth CMC hypersurfaces, with the same prescribed mean curvature, that converge to the given one. Moreover, these hypersurfaces arise as boundaries of minimisers. In ambient dimension $8$ the condition on the cone is redundant. (When the mean curvature vanishes identically, the result is the well-known Hardt--Simon approximation theorem.)

math.DG↗
arXiv2025

Extensions of Schoen--Simon--Yau and Schoen--Simon theorems via iteration à la De Giorgi

We give an alternative proof of the Schoen--Simon--Yau curvature estimates and associated Bernstein-type theorems (1975), and extend the original result by including the case of $6$-dimensional (stable minimal) immersions. The key step is an $ε$-regularity theorem, that assumes smallness of the scale-invariant $L^2$ norm of the second fundamental form. Further, we obtain a graph description, in the Lipschitz multi-valued sense, for any stable minimal immersion of dimension $n\geq 2$, that may have a singular set $Σ$ of locally finite $\mathcal{H}^{n-2}$-measure, and that is weakly close to a hyperplane. (In fact, if $\mathcal{H}^{n-2}(Σ)=0$, the conclusion is strengthened to a union of smooth graphs.) This follows directly from an $ε$-regularity theorem, that assumes smallness of the scale-invariant $L^2$ tilt-excess (verified when the hypersurface is weakly close to a hyperplane). Specialising the multi-valued decomposition to the case of embeddings, we recover the Schoen--Simon theorem (1981). In both $ε$-regularity theorems the relevant quantity (respectively, length of the second fundamental form and tilt function) solves a non-linear PDE on the immersed minimal hypersurface. The proof is carried out intrinsically (without linearising the PDE) by implementing an iteration method à la De Giorgi (from the linear De Giorgi--Nash--Moser theory). Stability implies estimates (intrinsic weak Caccioppoli inequalities) that make the iteration effective despite the non-linear framework. (In both $ε$-regularity theorems the method gives explicit constants that quantify the required smallness.)

math.AP↗
arXiv2022

Embeddedness of Min-Max CMC Hypersurfaces in Manifolds with Positive Ricci Curvature

We prove that on a compact Riemannian manifold of dimension $3$ or higher, with positive Ricci curvature, the Allen--Cahn min-max scheme (implemented by the first author and N. Wickramasekera in 2020), with prescribing function taken to be a non-zero constant $λ$, produces an embedded hypersurface of constant mean curvature $λ$ ($λ$-CMC). More precisely, we prove that the interface arising from said min-max contains no even-multiplicity minimal hypersurface and no quasi-embedded points (both of these occurrences are in principle possible in the conclusions of the aforementioned work by the first author and N. Wickramasekera). The immediate geometric corollary is the existence (in ambient manifolds as above) of embedded, closed $λ$-CMC hypersurfaces (with Morse index $1$) for any prescribed non-zero constant $λ$, with the expected singular set when the ambient dimension is $8$ or higher.

math.DG↗
arXiv2022

Hypersurfaces with mean curvature prescribed by an ambient function: compactness results

We consider, in a first instance, the class of boundaries of sets with locally finite perimeter whose (weakly defined) mean curvature is $g ν$, for a given continuous positive ambient function $g$, and where $ν$ denotes the inner normal. It is well-known that taking limits in the sense of varifolds within this class is not possible in general, due to the appearence of "hidden boundaries", that is, portions (of positive measure with even multiplicity) on which the (weakly defined) mean curvature vanishes, so that $g$ does not prescribe the mean curvature in the limit. As a special instance of a more general result, we prove that locally uniform $L^q$-bounds on the (weakly defined) second fundamental form, for $q>1$, in addition to the customary locally uniform bounds on the perimeters, lead to a compact class of boundaries with mean curvature prescribed by $g$. The proof relies on treating the boundaries as 'oriented integral varifolds', in order to exploit their orientability feature (that is lost when treating them as (unoriented) varifolds). Specifically, it relies on the formulation and analysis of a weak notion of curvature coefficients for oriented integral varifolds (inspired by Hutchinson's work). This framework gives (with no additional effort) a compactness result for oriented integral varifolds with curvature locally bounded in $L^q$ with $q>1$ and with mean curvature prescribed by any $g\in C^0$ (in fact, the function can vary with the varifold for which it prescribes the mean curvature, as long as there is locally uniform convergence of the prescribing functions). Our notions and statements are given in a Riemannian manifold, with the oriented varifolds that need not arise as boundaries (for instance, they could come from two-sided immersions).

math.DG↗
arXiv2021

Embeddedness of liquid-vapour interfaces in stable equilibrium

We consider a classical (capillary) model for a one-phase liquid in equilibrium. The liquid (e.g. water) is subject to a volume constraint, it does not mix with the surrounding vapour (e.g. air), it may come into contact with solid supports (e.g. a container), and is subject to the action of an analytic potential field (e.g. gravity). The region occupied by the liquid is described as a set of locally finite perimeter (Caccioppoli set) in $\mathbb{R}^3$; no a priori regularity assumption is made on its boundary. The (twofold) scope in this note is to propose a weakest possible set of mathematical assumptions that sensibly describe a condition of stable equilibrium for the liquid-vapour interface (the capillary surface), and to infer from those that this interface is a smoothly embedded analytic surface. (The liquid-solid-vapour junction, or free boundary, can be present but is not analysed here.) The result relies fundamentally on the recent varifold regularity theory developed by Wickramasekera and the author, and on the identification of a suitable formulation of the stability condition.

math.AP↗
arXiv2021

The inhomogeneous Allen--Cahn equation and the existence of prescribed-mean-curvature hypersurfaces

We prove that for any given compact Riemannian manifold $N$ of dimension $n+1 \geq 3$ and any non-negative Lipschitz function $g$ on $N$, there exists a quasi-embedded, boundaryless hypersurface $M \subset N,$ of class $C^{2, α}$ for any $α\in (0,1),$ such that $M$ is the image of a two-sided immersion whose mean curvature is given by $gν$ for an appropriate choice of continuous unit normal $ν$ to the immersion; and moreover, the singular set $Σ= \overline{M} \setminus M$ is empty if $2 \leq n \leq 6,$ finite if $n=7$ and satisfies ${\mathcal H}^{n-7 + γ}(Σ) = 0$ for every $γ>0$ if $n \geq 8$. Here quasi-embedded means that near every non-embedded point, $M$ is the union of two embedded $C^{2, α}$ disks intersecting tangentially with each disk lying on one side of the other. If $g >0$ then $\overline{M}$ is the boundary of a Caccioppoli set. Our proof of this theorem is PDE theoretic and relies, when $g>0$ and $g\in C^{1,1}(N)$, on (i) a mountain pass construction of solutions to the inhomogeneous Allen--Cahn equation and (ii) a regularity result for integral varifolds arising from a Morse-index bounded, energy bounded, sequence of solutions to the (inhomogeneous) Allen--Cahn equation. The case of non-negative Lipschitz $g$ follows by approximation, based on the estimates that we establish.

math.DG↗
arXiv2020

Generic existence of multiplicity-1 minmax minimal hypersurfaces via Allen--Cahn

In Guaraco's 2018 work a new proof was given of the existence of a closed minimal hypersurface in a compact Riemannian manifold $N^{n+1}$ with $n\geq 2$. This was achieved by employing an Allen--Cahn approximation scheme and a one-parameter minmax for the Allen--Cahn energy (relying on works by Hutchinson, Tonegawa, Wickramasekera to pass to the limit as the Allen-Cahn parameter tends to $0$). The minimal hypersurface obtained may a priori carry a locally constant integer multiplicity. Here we consider a minmax construction that is a modification of the one in the aforementioned work, by allowing an initial freedom on the choice of the valley points between which the mountain pass construction is carried out, and then optimising over said choice. We prove that, when $2\leq n\leq 6$ and the metric is bumpy, this minmax leads to a (smooth closed) minimal hypersurface with multiplicity $1$. (When $n=2$ this conclusion also follows from Chodosh--Mantoulidis's recent work.) As immediate corollary we obtain that every compact Riemannian manifold of dimension $n+1$, $2\leq n\leq 6$, endowed with a bumpy metric, admits a two-sided smooth closed minimal hypersurface (this existence conclusion also follows from Zhou's recent result for minmax constructions via Almgren--Pitts theory).

math.DG↗
arXiv2020

Stable prescribed-mean-curvature integral varifolds of codimension 1: regularity and compactness

In a previous paper we developed a regularity and compactness theory in Euclidean ambient spaces for codimension 1 weakly stable CMC integral varifolds satisfying two (necessary) structural conditions. Here we generalize this theory to the setting where the mean curvature (of the regular part of the varifold) is prescribed by a given $C^{1, α}$ ambient function $g$ and the ambient space is a general $(n+1)$-dimensional Riemannian manifold; we give general conditions that imply that the support of the varifold, away from a possible singular set of dimension $\leq n-7,$ is the image of a $C^{2}$ immersion with continuous unit normal $ν$ and mean curvature $gν$. These conditions also identify a compact class of varifolds subject to an additional uniform mass bound. If $g$ does not vanish anywhere, or more generally if the set $\{g=0\}$ is sufficiently small (e.g. if ${\mathcal H}^{n}\,(\{g=0\})=0$), then the results and their proofs are quite analogous to the CMC case. When $g$ is arbitrary however, a number of additional considerations must be taken into account: first, the correct second variation assumption is that of finite Morse index (rather than weak stability); secondly, to have a geometrically useful theory with compactness conclusions, one of the two structural conditions on the varifold must be weakened; thirdly, in view of easy examples, this weakening of a structural hypothesis necessitates additional hypotheses in order to reach the same conclusion. We identify two sets of additional assumptions under which this conclusion holds for general $g$: one of them gives regularity and compactness, the other one gives regularity and is applicable to certain phase transition problems. We also provide corollaries for multiplicity $1$ varifolds associated to reduced boundaries of Caccioppoli sets. In these cases, some or all of the structural assumptions become redundant.

math.DG↗
arXiv2020

Multiplicity-1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature

We address the one-parameter minmax construction, via Allen--Cahn energy, that has recently lead to a new proof of the existence of a closed minimal hypersurface in an arbitrary compact Riemannian manifold $N^{n+1}$ with $n\geq 2$ (see Guaraco's 2018 work). We obtain the following multiplicity-$1$ result: if the Ricci curvature of $N$ is positive then the minmax Allen--Cahn solutions concentrate around a multiplicity-$1$ hypersurface, that may have a singular set of dimension $\leq n-7$. This result is new for $n\geq 3$ (for $n=2$ it is also implied by the recent work by Chodosh--Mantoulidis). The argument developed here is geometric in flavour and exploits directly the minmax characterization of the solutions. An immediate corollary is that every compact Riemannian manifold $N^{n+1}$ with $n\geq 2$ and positive Ricci curvature admits a two-sided closed minimal hypersurface, possibly with a singular set of dimension at most $n-7$. This existence result also follows from multiplicity-$1$ results developed within the Almgren--Pitts framework, see works by Ketover-Marques-Neves, Zhou, Marques-Neves, Ramirez-Luna.

math.AP↗
arXiv2019

Sets with constant normal in Carnot groups: properties and examples

We analyze subsets of Carnot groups that have intrinsic constant normal, as they appear in the blowup study of sets that have finite sub-Riemannian perimeter. The purpose of this paper is threefold. First, we prove some mild regularity and structural results in arbitrary Carnot groups. Namely, we show that for every constant-normal set in a Carnot group its sub-Riemannian-Lebesgue representative is regularly open, contractible, and its topological boundary coincides with the reduced boundary and with the measure-theoretic boundary. We infer these properties from a cone property. Such a cone will be a semisubgroup with nonempty interior that is canonically associated with the normal direction. We characterize the constant-normal sets exactly as those that are arbitrary unions of translations of such semisubgroups. Second, making use of such a characterization, we provide some pathological examples in the specific case of the free-Carnot group of step 3 and rank 2. Namely, we construct a constant normal set that, with respect to any Riemannian metric, is not of locally finite perimeter; we also construct an example with non-unique intrinsic blowup at some point, showing that it has different upper and lower sub-Riemannian density at the origin. Third, we show that in Carnot groups of step 4 or less, every constant-normal set is intrinsically rectifiable, in the sense of Franchi, Serapioni, and Serra Cassano.

math.DG↗
arXiv2019

Curvature estimates and sheeting theorems for weakly stable CMC hypersurfaces

Weakly stable constant mean curvature (CMC) hypersurfaces are stable critical points of the area functional with respect to volume preserving deformations. We establish a pointwise curvature estimate (in the non-singular dimensions) and a sheeting theorem (in all dimensions) for weakly stable CMC hypersurfaces, giving an effective version of the compactness theorem for weakly stable CMC hypersurfaces established in the recent work of the first and third-named authors. Our results generalize the curvature estimate and the sheeting theorem proven respectively by Schoen--Simon--Yau and Schoen--Simon for strongly stable hypersurfaces.

math.DG↗
arXiv2018

Stable CMC integral varifolds of codimension $1$: regularity and compactness

We give two structural conditions on a codimension $1$ integral $n$-varifold with first variation locally summable to an exponent $p>n$ that imply the following: whenever each orientable portion of the $C^{1}$-embedded part of the varifold (which is non-empty by the Allard regularity theory) is stationarity and the $C^{2}$-immersed part of it is stable with respect to the area functional for volume preserving deformations, its support, except possibly on a closed set of codimension $7$, is an immersed constant-mean-curvature (cmc) hypersurface of class $C^{2}$ that can fail to be embedded only at points where locally the support is the union of two $C^{2}$ embedded cmc disks with only tangential intersection. Both structural conditions are necessary for the conclusions and involve only those parts of the varifold that are made up of embedded $C^{1, α}$-regular pieces coming together in a regular fashion, making them easy to check in principle. We show also that any family of codimension 1 integral varifolds satisfying these structural and variational hypotheses as well as locally uniform mass and mean curvature bounds is compact in the varifold topology. Our results generalize both the regularity theory of the second author (for stable minimal hypersurfaces) and the regularity theory of Schoen--Simon, for hypersurfaces satisfying a priori a smallness hypothesis on the singular set in addition to the variational hypotheses). Corollaries of the main varifold regularity theorem are obtained for sets of locally finite perimeter, which generalize the regularity theory of Gonzalez--Massari--Tamanini for boundaries that locally minimize perimeter subject to the fixed enclosed volume constraint.

math.DG↗
arXiv2015

Compactness results for triholomorphic maps

We consider triholomorphic maps from an almost hyper-Hermitian manifold $\mathcal{M}^{4m}$ into a hyperKähler manifold $\mathcal{N}^{4n}$. This means that $u \in W^{1,2}$ satisfies a quaternionic del-bar equation. We work under the assumption that $u$ is locally strongly approximable in $W^{1,2}$ by smooth maps: then such maps are almost stationary harmonic (when $\mathcal{M}$ is hyperKähler as well, then stationary harmonic). We show that in this more general situation the classical $ε$-regularity result still holds. We then address compactness issues for a weakly converging sequence $u_\ell \rightharpoonup u_\infty$ of strongly approximable triholomorphic maps $u_\ell:\mathcal{M} \to \mathcal{N}$ with uniformly bounded Dirichlet energies. The blow up analysis leads, as in the usual stationary setting, to the existence of a rectifiable blow-up set $Σ$ of codimension $2$, away from which the sequence converges strongly. The defect measure $Θ(x) {\mathcal{H}}^{4m-2} \llcorner Σ$ encodes the loss of energy in the limit; we prove that for a.e. point on $Σ$ the value of $Θ$ is given by the sum of energies of a (finite) number of smooth non-constant holomorphic bubbles (here the holomorphicity is understood w.r.t. a complex structure on $\mathcal{N}$ that depends on the chosen point on $Σ$). In the case that $\mathcal{M}$ is hyperKähler this result was established by C. Y. Wang (2003) with a different proof; we rely on Lorentz space estimates. By means of a calibration and a homological argument we further prove that for each portion of $Σ\setminus \text{Sing}_{u_\infty}$ contained in a Lipschitz graph we find a unique alm. compl. st. on $\mathcal{M}$ that makes the portion pseudoholomorphic and smooth, with $Θ$ constant; moreover the bubbles originating at points of such a smooth piece are holomorphic for a common complex structure.

math.AP↗
arXiv2015

Rate of decay for the mass ratio of pseudo-holomorphic integral 2-cycles

We consider any pseudo holomorphic integral 2-cycle in an arbitrary almost complex manifold and perform a blow up analysis at an arbitrary point. Building upon a pseudo algebraic blow up (previously introduced by the author) we prove a geometric rate of decay for the mass ratio towards the limiting density, with an explicit exponent of decay expressed in terms of the density of the current at the point.

math.AP↗
arXiv2012

Regularity of sets with constant horizontal normal in the Engel group

In the Engel group with its Carnot group structure we study subsets of locally finite subRiemannian perimeter and possessing constant subRiemannian normal. We prove the rectifiability of such sets: more precisely we show that, in some specific coordinates, they are upper-graphs of entire Lipschitz functions (with respect to the Euclidean distance). However we find that, when they are written as intrinsic upper-graphs with respect to the direction of the normal, then the function defining the set might even fail to be continuous. Nevertheless, we can prove that one can always find other horizontal directions for which the set is the upper-graph of a function that is Lipschitz-continuous with respect to the intrinsic distance (and in particular Hölder-continuous for the Euclidean distance). We further discuss a PDE characterization of the class of all sets with constant horizontal normal. Finally, we show that our rectifiability argument extends to the case of filiform groups of the first kind.

math.AP↗