The Journal of Geometric Analysis

LetM be a complete Riemannian manifold with Ricci curvature bounded from below. We give an explicit estimate for the size of the negative sets of solutions to the differential inequality Δu ≥λu where Δ is the Laplacian and λ is a negative constant. This inequality arises naturally when we study the lengthH of the mean curvature of an isometric immersionf of M into another Riemannian manifoldN with curvature bounded above by some constantκ. Suppose that the image f(M) does not meet the cut locus of some pointo ∈ N. As a consequence of our estimate, we prove that, givenρ > 0, if supH is less than a certain explicit expression μ(κ, ρ) inρ andκ on any domainU that contains an inscribed ball of radius greater than an explicitly computable numberR, then the diameter of the setf(U) inN must exceed 2ρ. Moreover, if supH = μ(κ, ρ) onM and the diameter off(M) inN equals 2ρ, thenf is a minimal immersion into a distance sphere of radiusρ inN.

In this paper, we consider holomorphic functions on the m-dimensional Lie ball LB(0, 1) which admit a square integrable extension on the Lie sphere. We then define orthogonal projections of this set onto suitable subsets of functions defined in lower- dimensional spaces to obtain several Radon-type transforms. For all these transforms we provide the kernel and an integral representation, besides other properties. In particular, we introduce and study a generalization to the Lie ball of the Szegő–Radon transform introduced in Colombo et al. (Adv Appl Math 74:1–22, 2016), and various types of Hua–Radon transforms.

We study a characterization of 4-dimensional (not necessarily complete) gradient Ricci solitons (M, g, f) which have harmonic Weyl curvature, i.e., $$\delta W=0$$ . Roughly speaking, we prove that the soliton metric g is locally isometric to one of the following four types: an Einstein metric, the product $$ \mathbb {R}^2 \times N_{\lambda }$$ of the Euclidean metric and a 2-d Riemannian manifold of constant curvature $${\lambda } \ne 0$$ , a certain singular metric and a locally conformally flat metric. The method here is motivated by Cao–Chen’s works (in Trans Am Math Soc 364:2377–2391, 2012; Duke Math J 162:1003–1204, 2013) and Derdziński’s study on Codazzi tensors (in Math Z 172:273–280, 1980). Combined with the previous results on locally conformally flat solitons, our characterization yields a new classification of 4-d complete steady solitons with $$\delta W=0$$ . For the shrinking case, it re-proves the rigidity result (Fernández-López and García-Río in Math Z 269:461–466, 2011; Munteanu and Sesum in J. Geom Anal 23:539–561, 2013) in 4-d. It also helps to understand the expanding case; we now understand all 4-d non-conformally flat ones with $$\delta W=0$$ . We also characterize locally 4-d (not necessarily complete) gradient Ricci solitons with harmonic curvature.

Let ƒ be a polynomial automorphism of ℂk of degree λ, whose rational extension to ℙk maps the hyperplane at infinity to a single point. Given any positive closed current S on ℙk of bidegree (1,1), we show that the sequence λ−n(ƒn)*S converges in the sense of currents on ℙk to a linear combination of the Green current T+ of ƒ and the current of integration along the hyperplane at infinity. We give an interpretation of the coefficients in terms of generalized Lelong numbers with respect to an invariant dynamical current for ƒ−1.

We prove that every closed, smooth $$n$$ -manifold $$X$$ admits a Riemannian metric together with a constant mean curvature (CMC) foliation if and only if its Euler characteristic is zero, where by a CMC foliation we mean a smooth, codimension-one, transversely oriented foliation with leaves of CMC and where the value of the CMC can vary from leaf to leaf. Furthermore, we prove that this CMC foliation of $$X$$ can be chosen so that when $$n\ge 2$$ , the constant values of the mean curvatures of its leaves change sign. We also prove a general structure theorem for any such non-minimal CMC foliation of $$X$$ that describes relationships between the geometry and topology of the leaves, including the property that there exist compact leaves for every attained value of the mean curvature.

We study the existence of left invariant closed $$G_2$$ -structures defining a Ricci soliton metric on simply connected nonabelian nilpotent Lie groups. For each one of these $$G_2$$ -structures, we show long time existence and uniqueness of solution for the Laplacian flow on the noncompact manifold. Moreover, considering the Laplacian flow on the associated Lie algebra as a bracket flow on $${\mathbb {R}}^7$$ in a similar way as in Lauret (Commun Anal Geom 19(5):831–854, 2011) we prove that the underlying metrics $$g(t)$$ of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric, uniformly on compact sets in the nilpotent Lie group, as $$t$$ goes to infinity.

We prove finite jet determination results for smooth CR embeddings, which are of constant degeneracy, using the method of complete systems. As an application, we obtain a reflection principle for mappings between a Levinondegenerate hypersurface in ℂ N and a Levinondegenerate hypersurface in ℂN+1.We also give an independent proof of the reflection principle for mappings between strictly pseudoconvex hypersurfaces in any codimension due to Forstneric [14].

We prove a Tb theorem on quasimetric spaces equipped with what we call an upper doubling measure. This is a property that encompasses both the doubling measures and those satisfying the upper power bound μ(B(x,r))≤Cr d . Our spaces are only assumed to satisfy the geometric doubling property: every ball of radius r can be covered by at most N balls of radius r/2. A key ingredient is the construction of random systems of dyadic cubes in such spaces.