In this Note we prove that, under a weaker condition than the-lemma, the existence of balanced metrics is preserved under small deformations. This weaker condition is satisfied on the twistor space over a compact self-dual four manifold.

Let M be a compact n-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary ∂M. Assume that the mean curvature H of the boundary ∂M satisfies H≥(n−1)k>0 for some positive constant k. In this paper, we prove that the distance function d to the boundary ∂M is bounded from above by 1/𝑘 and the upper bound is achieved if and only if M is isometric to an n-dimensional Euclidean ball of radius 1/𝑘.

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In this article, we define a new class of middle dimensional submanifolds of a Hyperkahler manifold which contains the class of complex Lagrangian submanifolds, and show that this larger class is invariant under the mean curvature flow. Along the flow, the complex phase map satisfies the generalized harmonic map heat equation, tt is also related to the mean curvature vector via a first order differential equation. Moreover, we proved a result on nonexistence of Type I singularity.

Using deformations of singular twistor spaces, a generalisation of the connected sum construction appropriate for quaternionic manifolds is introduced. This is used to construct examples of quaternionic manifolds which have no quaternionic symmetries and leads to examples of quaternionic manifolds whose twistor spaces have arbitrary algebraic dimension.