Rigidity of generalized laplacians and some geometric applications.
Rigidity of noncompact manifolds with cyclic parallel Ricci curvature
We prove that if M is a complete noncompact Riemannian manifold whose Ricci tensor is cyclic parallel and whose scalar curvature is nonpositive, then M is Einstein, provided the Sobolev constant is positive and an integral inequality is satisfied.
Rough isometries and p-harmonic functions with finite Dirichlet integral.
Scalar curvature and connected sums of self-dual 4-manifolds
Under a reasonable vanishing hypothesis, Donaldson and Friedman proved that the connected sum of two self-dual Riemannian 4-manifolds is again self-dual. Here we prove that the same result can be extended to the positive scalar curvature case. This is an analogue of the classical theorem of Gromov–Lawson and Schoen–Yau in the self-dual category. The proof is based on twistor theory.
Scalar curvature, covering spaces, and Seiberg-Witten theory.
Schiffer problem and isoparametric hypersurfaces.
The Schiffer Problem as originally stated for Euclidean spaces (and later for some symmetric spaces) is the following: Given a bounded connected open set Ω with a regular boundary and such that the complement of its closure is connected, does the existence of a solution to the Overdetermined Neumann Problem (N) imply that Ω is a ball? The same question for the Overdetermined Dirichlet Problem (D). We consider the generalization of the Schiffer problem to an arbitrary Riemannian manifold and also...
Schouten tensor equations in conformal geometry with prescribed boundary metric.
Second order parallel tensors on generalized Sasakian space forms and semi parallel hypersurfaces in Sasakian space forms.
Second order parallel tensors on -contact metric manifolds.
Semi-invariant submanifolds of Riemannian product manifold.
Sharp bounds for the intersection of nodal lines with certain curves
Let be a hyperbolic surface and let be a Laplacian eigenfunction having eigenvalue with . Let be the set of nodal lines of . For a fixed analytic curve of finite length, we study the number of intersections between and in terms of . When is compact and a geodesic circle, or when has finite volume and is a closed horocycle, we prove that is “good” in the sense of [TZ]. As a result, we obtain that the number of intersections between and is . This bound is sharp.
Sharp isoperimetric inequalities and model spaces for the Curvature-Dimension-Diameter condition
We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex support are bounded from above (possibly infinitely). Our inequalities are sharp for sets of any given measure and with respect to all parameters (curvature, dimension and diameter). Moreover, for each choice of parameters, we identify the model spaces which...
Small eigenvalues on Y-pieces and on Riemann surfaces.
Small time heat kernel behavior on Riemannian complexes.
Smoothing Riemannian metrics with Ricci curvature bounds.
Sobolev-Kantorovich Inequalities
In a recent work, E. Cinti and F. Otto established some new interpolation inequalities in the study of pattern formation, bounding the Lr(μ)-norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich-Wasserstein distance to μ. This article emphasizes this family of interpolation inequalities, called Sobolev-Kantorovich inequalities, which may be established in the rather large setting of non-negatively curved (weighted) Riemannian manifolds by means...
Solutions of the constraint equations in general relativity satisfying "hyperboloidal boundary conditions" [Book]
Solving -Laplacian equations on complete manifolds.
Some Additive Designs
Given a finite additive abelian group and an integer , with , denote by the simple incidence structure whose point-set is and whose blocks are the -subsets of such that . It is known (see [Caggegi, A., Di Bartolo, A., Falcone, G.: Boolean 2-designs and the embedding of a 2-design in a group arxiv 0806.3433v2, (2008), 1–8.]) that is a 2-design, if is an elementary abelian -group with a prime divisor of . From [Caggegi, A., Falcone, G., Pavone, M.: On the additivity of block...
Some applications of Ricci flow to 3-manifolds