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.

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...

Let $Y$ be a hyperbolic surface and let $\phi$ be a Laplacian eigenfunction having eigenvalue $-1/4-{\tau }^2$ with $\tau >0$. Let $N\left(\phi \right)$ be the set of nodal lines of $\phi$. For a fixed analytic curve $\gamma$ of finite length, we study the number of intersections between $N\left(\phi \right)$ and $\gamma$ in terms of $\tau$. When $Y$ is compact and $\gamma$ a geodesic circle, or when $Y$ has finite volume and $\gamma$ is a closed horocycle, we prove that $\gamma$ is “good” in the sense of [TZ]. As a result, we obtain that the number of intersections between $N\left(\phi \right)$ and $\gamma$ is $O\left(\tau \right)$. This bound is sharp.

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...

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...

Given a finite additive abelian group $G$ and an integer $k$, with $3\le k\le \left|G\right|$, denote by ${𝒟}_k\left(G\right)$ the simple incidence structure whose point-set is $G$ and whose blocks are the $k$-subsets $C=\{c_1,c_2,\cdots ,c_k\}$ of $G$ such that $c_1+c_2+\cdots +c_k=0$. 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 ${𝒟}_k\left(G\right)$ is a 2-design, if $G$ is an elementary abelian $p$-group with $p$ a prime divisor of $k$. From [Caggegi, A., Falcone, G., Pavone, M.: On the additivity of block...

In this paper some properties of an immersion of two-dimensional surface with boundary into $Esp4$ are studied. The main tool is the maximal principle property of a solution of the elliptic system of partial differential equations. Some conditions for a surface to be a part of a 2-dimensional spheren in $Esp4$ are presented.

In this paper, we consider some evolution equations of generalized Ricci curvature and generalized scalar curvature under the List’s flow. As applications, we obtain $L^2$-estimates for generalized scalar curvature and the first variational formulae for non-negative eigenvalues with respect to the Laplacian.

We prove existence of positive solutions for the equation $-{\mathrm{△}}_{g_0}u+u=\left(1+ϵK\left(x\right)\right)u^{2^{*}-1}$ on $S^n$, arising in the prescribed scalar curvature problem. is the Laplace-Beltrami operator on $S^n$, $2^{\mathrm{\ast }}$ is the critical Sobolev exponent, and $ϵ$ is a small parameter. The problem can be reduced to a finite dimensional study which is performed with Morse theory.