Motivated by M\"uller-Haslhofer results on the dynamical stability and instability of Ricci-flat metrics under Ricci flow, we obtain for pairs vanishing 3-forms generalized flow.

We define a class of two dimensional surfaces conformally related to minimal surfaces in flat three dimensional geometries. By the utility of the metrics of such surfaces we give a construction of the metrics of 2N dimensional Ricci flat (pseudo-) Riemannian geometries.

We develop numerical algorithms for solving the Einstein equation on Calabi-Yau manifolds at arbitrary values of their complex structure and Kähler parameters. We show that Kähler geometry can be exploited for significant gains in computational efficiency. As a proof of principle, we apply our methods to a one-parameter family of K3 surfaces constructed as blow-ups of the T /Z2 orbifold with ma...

In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations when the manifold is locally conformally flat or the Ricci curvature is positive. Dedicated to the memory of Profess...

We show that two of Bryant-Salamon’s G2-manifolds have a simple topology, S \ S or S \ CP . In this connection, we show there exists a complete Ricci-flat (non-flat) metric on Sn \ Sm for some n − 1 > m. We also give many examples of special Lagrangian submanifolds of T ∗Sn with the Stenzel metric. The hypersurface geometry is essential in the argument.

We study the stability and instability of ALE Ricci-flat metrics around which a Łojasiewicz inequality is satisfied for variation Perelman’s $$\lambda $$ functional adapted to situation denoted _{{\text {ALE}}}$$ . This was introduced by authors in recent work it has been proven that satisfies good enough at least neighborhoods integrable dimension larger than or equal 5.

We show that two of the Bryant-Salamon G2-manifolds have a simple topology, S \S or S \CP . In this connection, we show there exists a complete Ricci-flat (non-flat) metric on Sn \ Sm for some n − 1 > m. We also give many examples of special Lagrangian submanifolds of T ∗Sn with the Stenzel metric. Hypersurface geometry is essential for these arguments.