For any n-dimensional smooth manifold $\Sigma$, we show that all the singularities of the mean curvature flow with any initial mean convex hypersurface in $\Sigma$ are cylindrical (of convex type) if the flow converges to a smooth hypersurface $M_\infty$ (maybe empty) at infinity. Previously this was shown (i) for n$\leq$7, and (ii) for arbitrary n up to the first singular time without the smooth condition on $M_\infty$.

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Motivated by new explicit positive Ricci curvature metrics on the four-sphere which are also Einstein-Weyl, we show that the dimension of the Einstein-Weyl moduli near certain Einstein metrics is bounded by the rank of the isometry group and that any Weyl manifold can be embedded as a hypersurface with prescribed second fundamental form in some Einstein-Weyl space. Closed four-dimensional Einstein-Weyl manifolds are proved to be absolute minima of the L²-norm of the curvature of Weyl manifolds and a local version of the Lafontaine inequality is obtained. The above metrics on the four-sphere are shown to contain minimal hypersurfaces isometric to S¹S² whose second fundamental form has constant length.

Many years ago, SY Cheng, P. Li and myself [1] gave an estimate of the heat kernel for the Laplacian. This was later improved by P. Li and myself in [2]. The key point of this later paper was a parabolic Harnack inequality that generalized an elliptic Harnack inequality that I developed more than twenty years ago.

Motivated by the study of coupled Kähler-Einstein metrics by Hultgren and Witt Nyström (2018) and coupled Kähler-Ricci solitons by Hultgren (2017), we study in this paper coupled Sasaki-Einstein metrics and coupled Sasaki-Ricci solitons. We first show an isomorphism between the Lie algebra of all transverse holomorphic vector fields and certain space of coupled basic functions related to coupled twisted Laplacians for basic functions, and obtain extensions of the well-known obstructions to the existence of Kähler-Einstein metrics to this coupled case. These results are reduced to coupled Kähler-Einstein metrics when the Sasaki structure is regular. Secondly we show the existence of toric coupled Sasaki-Einstein metrics when the basic first Chern class is positive extending the work of Hultgren (2017).