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We show that in a closed 3-manifold with a generic metric of positive Ricci curvature, there are minimal surfaces of arbitrary large Morse index, which partially confirms a conjecture by Marques and Neves. We prove this by analyzing the lamination structure of the limit of minimal surfaces with bounded Morse index.

In this paper, we classify n-dimensional (n ≥ 3) complete non-compact locally conformally flat gradient steady solitons. In particular, we prove that a complete noncompact nonflat locally conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton.

Let X CPN be a smooth subvariety. We study a flow, called balancing flow, on the space of projectively equivalent embeddings of X which attempts to deform the given embedding into a balanced one. If L X is an ample line bundle, considering embeddings via H0(Lk) gives a sequence of balancing flows. We prove that, provided these flows are started at appropriate points, they converge to Calabi flow for as long as it exists. This result is the parabolic analogue of Donaldsons theorem relating balanced embeddings to metrics with constant scalar curvature [12]. In our proof we combine Donaldson乫s techniques with an asymptotic result of Liu and Ma [17] which, as we explain, describes the asymptotic behavior of the derivative of the map FS . Hilb whose fixed points are balanced metrics.

H. Hopf showed that the only constant mean curvature sphere S2 immersed in R3 is the round sphere. The K¨ahler framework is an adequate approach to generalize Hopf’ s theorem to higher dimensions. When ϕ : M → Rn is an isometric immersion from a K¨ahler manifold, the complexified second fundamental form α splits according to types. The (1, 1) part of the second fundamental form plays the role of the mean curvature for surfaces and will be called the pluri-mean curvature pmc. Therefore isometric immersions with parallel pluri-mean curvature (ppmc isometric immersions) generalize in a natural way the cmc immersions. It is a standard fact that R8 is the smallest space where CP2 can be embedded. The aim of this work is to generalize Hopf’s theorem proving in particular that the only ppmc isometric immersion from CP2 into R8 is the standard immersion.