Annals of Global Analysis and Geometry最新文献

We present a construction of closed 7-manifolds of holonomy (G_2), which generalises Kovalev’s twisted connected sums by taking quotients of the pieces in the construction before gluing. This makes it possible to realise a wider range of topological types, and Crowley, Goette and the author use this to exhibit examples of closed 7-manifolds with disconnected moduli space of holonomy (G_2) metrics.

We solve a super Toda system on a closed Riemann surface of genus (gamma >1) and with some particular spin structures. This generalizes the min–max methods and results for super Liouville equations and gives new existence results for super Toda systems.

In 2012, M. M. Graev associated to a compact homogeneous space G/H a nerve ({text {X}}_{G/H}), whose non-contractibility implies the existence of a G-invariant Einstein metric on G/H. The nerve ({text {X}}_{G/H}) is a compact, semi-algebraic set, defined purely Lie theoretically by intermediate subgroups. In this paper we present a detailed description of the work of Graev and the curvature estimates given by Böhm in 2004.

If one thinks of a Riemannian metric, (g_1), analogously as the gradient of the corresponding distance function, (d_1), with respect to a background Riemannian metric, (g_0), then a natural question arises as to whether a corresponding theory of Sobolev inequalities exists between the Riemannian metric and its distance function. In this paper, we study the sub-critical case (p < frac{m}{2}) where we show a Sobolev inequality exists between a Riemannian metric and its distance function. In particular, we show that an (L^{frac{p}{2}}) bound on a Riemannian metric implies an (L^q) bound on its corresponding distance function. We then use this result to state a convergence theorem and show how this theorem can be useful to prove geometric stability results by proving a version of Gromov’s conjecture for tori with almost non-negative scalar curvature in the conformal case. Examples are given to show that the hypotheses of the main theorems are necessary.

In this paper, we extend existing results for generalized solitons, called q-solitons, to the complete case by considering non-compact solitons. By placing regularity conditions on the vector field X and curvature conditions on M, we are able to use the chosen properties of the tensor q to see that such non-compact q-solitons are stationary and q-flat. We conclude by applying our results to the examples of ambient obstruction solitons, Cotton solitons, and Bach solitons to demonstrate the utility of these general theorems for various flows.

The Euclidean unit sphere in dimension n minimizes the first positive eigenvalue of the Laplacian among all the compact, Riemannian manifolds of dimension n with Ricci curvature bounded below by (n-1) as a consequence of Lichnerowicz’s theorem. The eigenspectrum of the Laplacian is given by a non-decreasing sequence of real numbers tending to infinity. In dimension two, we prove that such an inequality holds for the subsequent eigenvalues in the sequence for ellipsoids that are obtained as analytic perturbations of the Euclidean unit sphere for the truncated spectrum.

We provide a new cohomological obstruction to the existence of astheno-Kähler metrics on compact complex manifolds. Several results of independent interests regarding the Bott-Chern and Aeppli cohomology groups are presented and relevant examples are discussed.

The aim of this paper is to construct left-invariant Einstein pseudo-Riemannian Sasaki metrics on solvable Lie groups. We consider the class of (mathfrak {z})-standard Sasaki solvable Lie algebras of dimension (2n+3), which are in one-to-one correspondence with pseudo-Kähler nilpotent Lie algebras of dimension 2n endowed with a compatible derivation, in a suitable sense. We characterize the pseudo-Kähler structures and derivations giving rise to Sasaki–Einstein metrics. We classify (mathfrak {z})-standard Sasaki solvable Lie algebras of dimension (le 7) and those whose pseudo-Kähler reduction is an abelian Lie algebra. The Einstein metrics we obtain are standard, but not of pseudo-Iwasawa type.

The goal of this paper is to study Yamabe flow on a complete Riemannian manifold of bounded geometry with possibly infinite volume. In case of infinite volume, standard volume normalization of the Yamabe flow fails and the flow may not converge. Instead, we consider a curvature normalized Yamabe flow, and assuming negative scalar curvature, prove its long-time existence and convergence. This extends the results of Suárez-Serrato and Tapie to a non-compact setting. In the appendix we specify our analysis to a particular example of manifolds with bounded geometry, namely manifolds with fibered boundary metric. In this case we obtain stronger estimates for the short time solution using microlocal methods.