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Michele Rimoldi

Publications and source records attributed to Michele Rimoldi.

At least 19 recordsLinked to original sources
arXiv2026

Sobolev and Michael-Simon inequalities via the ABP method beyond Euclidean volume growth

We develop an ABP approach to Sobolev and Michael-Simon type inequalities under volume noncollapsing assumptions. The main new observation is a refinement of Brendle's contact-set argument: the ABP image contains the full geodesic ball centered at the minimum point of the Neumann potential, with radius equal to the ABP parameter. This allows one to use lower bounds for the volumes of geodesic balls, either at a fixed scale or under prescribed volume-growth assumptions, rather than positive asymptotic volume ratio. The central application is a Michael-Simon type inequality for immersed submanifolds of ambient manifolds with nonnegative sectional curvature and volume noncollapsing. The resulting inequality contains a lower-order term determined by the noncollapsing scale and applies to submanifolds with controlled mean curvature. In the intrinsic case, the same method gives an ABP proof of Varopoulos' $L^{1}$-Sobolev inequality with lower-order term, identifying the optimal constant in front of the gradient term, as well as explicit lower bounds for the isoperimetric profile in terms of lower bounds on the volumes of geodesic balls. Further geometric applications include Topping-type diameter estimates for submanifolds involving the $L^{n-1}$-norm of the mean curvature and various heat kernel and spectral estimates in the minimal case.

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arXiv2026

Classification of quadratically pinched self-shrinkers in higher codimension

We classify properly immersed self-shrinkers of the mean curvature flow in arbitrary codimension under a quadratic pinching condition of Andrews-Baker type on the second fundamental form that is preserved along the flow. Under this assumption, such self-shrinkers reduce effectively to codimension one and are therefore generalized self-shrinking cylinders. In contrast to previous works, our approach is purely elliptic: it relies on parabolicity in a weighted setting and is tailored specifically to self-shrinkers, rather than to general ancient solutions of the flow. This allows us to avoid assuming any uniform pinching condition, to treat in any dimension the sharp Andrews-Baker pinching constant $\frac{4}{3n}$ and hence to sharpen, in the self-shrinker setting, the pinching constants appearing in recent classification results for ancient solutions.

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arXiv2025

Asymptotically non-negative Ricci curvature, elliptic Kato constant and isoperimetric inequalities

The ABP method for proving isoperimetric inequalities has been first employed by Cabré in $\mathbb{R}^n$, then developed by Brendle, notably in the context of non-compact Riemannian manifolds of non-negative Ricci curvature and positive asymptotic volume ratio. In this paper, we expand upon their approach and prove isoperimetric inequalities (sharp in the limit) in the presence of a small amount of negative curvature. First, we consider smallness of the negative part $\mathrm{Ric}_-$ of the Ricci curvature in terms of its elliptic Kato constant. Indeed, the Kato constant turns out to control the non-negativity of the ($\infty$-)Bakry-Émery Ricci-tensor of a suitable conformal deformation of the manifold, and the ABP method can be implemented in this setting. Secondly, we show that the smallness of the Kato constant is ensured provided that the asymptotic volume ratio is positive and either $M$ has one end and asymptotically non-negative sectional curvature, or there is a suitable polynomial decay of $\mathrm{Ric}_{-}$, and the relative volume comparison condition known as $\textbf{(VC)}$ holds. To show this latter fact, we enhance techniques elaborated by Li-Tam and Kasue to obtain new estimates of the Green function valid on the whole manifold.

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arXiv2025

Rigidity and non-existence results for collapsed translators

We prove a rigidity result for mean curvature self-translating solitons, characterizing the grim reaper cylinder as the only finite entropy self-translating 2-surface in $\mathbb{R}^3$ of width $π$ and bounded from below. The proof makes use of parabolicity in a weighted setting applied to a suitable universally $L$-superharmonic function defined on translaters in such slabs.

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arXiv2025

Isoperimetric and Michael-Simon inequalities on manifolds with asymptotically nonnegative curvature

We establish the validity of the isoperimetric inequality (or equivalently, an $L^1$ Euclidean-type Sobolev inequality) on manifolds with asymptotically non-negative sectional curvature. Unlike previous results in the literature, our approach does not require the negative part of the curvature to be globally small. Furthermore, we derive a Michael-Simon inequality on manifolds whose curvature is non-negative outside a compact set. The proofs employ the ABP method for isoperimetry, initially introduced by Cabr\'e in the Euclidean setting and subsequently extended and skillfully adapted by Brendle to the challenging context of non-negatively curved manifolds. Notably, we show that this technique can be localized to appropriate regions of the manifold. Additional key elements of the argument include the geometric structure at infinity of asymptotically non-negatively curved manifolds, their spectral properties - which ensure the non-negativity of a Bakry-\'Emery Ricci tensor on a conformal deformation of each end - and a result that deduces the validity of the isoperimetric inequality on the entire manifold, provided it holds outside a compact set.

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arXiv2023

Poincaré inequality and topological rigidity of translators and self-expanders for the mean curvature flow

We prove an abstract structure theorem for weighted manifolds supporting a weighted $f$-Poincaré inequality and whose ends satisfy a suitable non-integrability condition. We then study how our arguments can be used to obtain full topological control on two important classes of hypersurfaces of the Euclidean space, namely translators and self-expanders for the mean curvature flow, under either stability or curvature asumptions. As an important intermediate step in order to get our results we get the validity of a Poincaré inequality with respect to the natural weighted measure on any translator and we prove that any end of a translator must have infinite weighted volume. Similar tools can be obtained for properly immersed self-expanders permitting to get topological rigidity under curvature assumptions.

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arXiv2021

Density and non-density of $C^\infty_c \hookrightarrow W^{k,p}$ on complete manifolds with curvature bounds

We investigate the density of compactly supported smooth functions in the Sobolev space $W^{k,p}$ on complete Riemannian manifolds. In the first part of the paper, we extend to the full range $p\in [1,2]$ the most general results known in the Hilbertian case. In particular, we obtain the density under a quadratic Ricci lower bound (when $k=2$) or a suitably controlled growth of the derivatives of the Riemann curvature tensor only up to order $k-3$ (when $k>2$). To this end, we prove a gradient regularity lemma that might be of independent interest. In the second part of the paper, for every $n \ge 2$ and $p>2$ we construct a complete $n$-dimensional manifold with sectional curvature bounded from below by a negative constant, for which the density property in $W^{k,p}$ does not hold for any $k \ge 2$. We also deduce the existence of a counterexample to the validity of the Calderón-Zygmund inequality for $p>2$ when $\mathrm{Sec} \ge 0$, and in the compact setting we show the impossibility to build a Calderón-Zygmund theory for $p>2$ with constants only depending on a bound on the diameter and a lower bound on the sectional curvature.

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arXiv2020

Higher order distance-like functions and Sobolev spaces

We consider complete Riemannian manifolds with a controlled growth of the covariant derivatives of Ricci curvatures up to order $k-2$ and a controlled decay of the injectivity radii. On such manifolds we construct distance-like functions with a control on covariant derivatives up to order $k$. Alternatively, the assumption on the injectivity radii can be replaced with the request of a controlled growth of the full curvature tensor at order $0$. The control in the assumptions occur via non-necessarily polynomial growth functions. This construction largely extend previously known results in various directions, permitting to obtain consequences which are (in a sense) sharp. A first main application is to the study of the density property for Sobolev spaces on Riemannian manifolds, namely the problem of guaranteeing the density of smooth compactly supported function in the Sobolev space $W^{k,p}$. Contrary to all previously known results this can be obtained also on manifolds with possibly unbounded geometry. In the particular case $p=2$, making use of the Weitzenböck formula for a Lichnerowicz Laplacian acting on the space of smooth section of the bundle of $k$-covariant symmetric tensors, we can weaken the assumptions needed to obtain the density property. Namely we prove that the control on the highest order derivative of curvature is not needed in this situation. Beyond the density property we finally highlight some new applications of our results to disturbed Sobolev inequalities, disturbed $L^{p}$-Calderón-Zygmund inequalities and the full Omori-Yau maximum principle for the Hessian.

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arXiv2020

The Frankel property for self-shrinkers from the viewpoint of elliptic PDE's

We show that two properly embedded self-shrinkers in Euclidean space that are sufficiently separated at infinity must intersect at a finite point. The proof is based on a localized version of the Reilly formula applied to a suitable f-harmonic function with controlled gradient. In the immersed case, a new direct proof of the generalized half-space property is also presented.

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arXiv2019

Density problems for second order Sobolev spaces and cut-off functions on manifolds with unbounded geometry

We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradient and mild growth of the Hessian. As a main application, we prove that smooth compactly supported functions are dense in $W^{2,p}$. The result is improved for $p=2$ avoiding both the upper bound on the Ricci tensor, and the injectivity radius assumption. As further applications we prove new disturbed Sobolev and Calderón-Zygmund inequalities on manifolds with possibly unbounded curvature and highlight consequences about the validity of the full Omori-Yau maximum principle for the Hessian.

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arXiv2019

Extremals of Log Sobolev inequality on non-compact manifolds and Ricci soliton structures

In this paper we establish the existence of extremals for the Log Sobolev functional on complete non-compact manifolds with Ricci curvature bounded from below and strictly positive injectivity radius, under a condition near infinity. When Ricci curvature is also bounded from above we get exponential decay at infinity of the extremals. As a consequence of these analytical results we establish, under the same assumptions, that non-trivial shrinking Ricci solitons support a gradient Ricci soliton structure. On the way, we prove two results of independent interest: the existence of a distance-like function with uniformly controlled gradient and Hessian on complete non-compact manifolds with bounded Ricci curvature and strictly positive injectivity radius and a general growth estimate for the norm of the soliton vector field on manifolds with bounded Ricci curvature.

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arXiv2019

Quantitative index bounds for translators via topology

We obtain a quantitative estimate on the generalised index of translators for the mean curvature flow with bounded norm of the second fundamental form. The estimate involves the dimension of the space of weighted square integrable f-harmonic 1-forms. By the adaptation to the weighted setting of Li-Tam theory developed in previous works, this yields estimates in terms of the number of ends of the hypersurface when this is contained in a upper halfspace with respect to the translating direction. When there exists a point where all principal curvatures are distinct we estimate the nullity of the stability operator. This permits to obtain quantitative estimates on the stability index via the topology of translators with bounded norm of the second fundamental form which are either two-dimensional or (in higher dimension) have finite topological type and are contained in a upper halfspace.

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arXiv2018

Index and first Betti number of f-minimal hypersurfaces: general ambients

We generalize a method by L. Ambrozio, A. Carlotto, and B. Sharp to study the Morse index of closed f-minimal hypersurfaces isometrically immersed in a general weighted manifold. The technique permits, in particular, to obtain a linear lower bound on the Morse index via the first Betti number for closed f-minimal hypersurfaces in products of some compact rank one symmetric spaces with an Euclidean factor, endowed with the rigid shrinking gradient Ricci soliton structure. These include, as particular cases, all cylindric shrinking gradient Ricci solitons.

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arXiv2018

Index and first Betti number of $f$-minimal hypersurfaces and self-shrinkers

We study the Morse index of self-shrinkers for the mean curvature flow and, more generally, of $f$-minimal hypersurfaces in a weighted Euclidean space endowed with a convex weight. When the hypersurface is compact, we show that the index is bounded from below by an affine function of its first Betti number. When the first Betti number is large, this improves index estimates known in literature. In the complete non-compact case, the lower bound is in terms of the dimension of the space of weighted square summable $f$-harmonic $1$-forms; in particular, in dimension $2$, the procedure gives an index estimate in terms of the genus of the surface.

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arXiv2014

Stability properties and topology at infinity of f-minimal hypersurfaces

We study stability properties of $f$-minimal hypersurfaces isometrically immersed in weighted manifolds with non-negative Bakry-Emery Ricci curvature under volume growth conditions. Moreover, exploiting a weighted version of a finiteness result and the adaptation to this setting of Li-Tam theory, we investigate the topology at infinity of $f$-minimal hypersurfaces. On the way, we prove a new comparison result in weighted geometry and we provide a general weighted $L^1$-Sobolev inequality for hypersurfaces in Cartan-Hadamard weighted manifolds, satisfying suitable restrictions on the weight function.

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arXiv2014

The Cotton tensor and the Ricci flow

We compute the evolution equation of the Cotton and the Bach tensor under the Ricci flow of a Riemannian manifold, with particular attention to the three dimensional case, and we discuss some applications.

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arXiv2013

Topology of steady and expanding gradient Ricci solitons via f-harmonic maps

In this paper we give some results on the topology of manifolds with $\infty$-Bakry-Émery Ricci tensor bounded below, and in particular of steady and expanding gradient Ricci solitons. To this aim we clarify and further develop the theory of f-harmonic maps from non-compact manifolds into non-positively curved manifolds. Notably, we prove existence and vanishing results which generalize to the weighted setting part of Schoen and Yau's theory of harmonic maps.

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