Annals of Global Analysis and Geometry最新文献

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.

A smooth closed manifold M is called almost Ricci-flat if

where (text {Ric}_g) and (text {diam}_g), respectively, denote the Ricci tensor and the diameter of g and g runs over all Riemannian metrics on M. By using Kummer-type method, we construct a smooth closed almost Ricci-flat nonspin 5-manifold M which is simply connected. It is minimal volume vanishes; namely, it collapses with sectional curvature bounded.

We investigate the Lax equation in the context of infinite-dimensional Lie algebras. Explicit solutions are discussed in the sequentially complete asymptotic estimate context, and an integral expansion (sums of iterated Riemann integrals over nested commutators with correction term) is derived for the situation that the Lie algebra is inherited by an infinite-dimensional Lie group in Milnor’s sense. In the context of Banach Lie groups (and Lie groups with suitable regularity properties), we generalize the Baker–Campbell–Dynkin–Hausdorff formula to the product integral (with additional nilpotency assumption in the non-Banach case). We combine this formula with the results obtained for the Lax equation to derive an explicit representation of the product integral in terms of the exponential map. An important ingredient in the non-Banach case is an integral transformation that we introduce. This transformation maps continuous Lie algebra-valued curves to smooth ones and leaves the product integral invariant. This transformation is also used to prove a regularity statement in the asymptotic estimate context.

This article studies Kummer K3 surfaces close to the orbifold limit. We improve upon estimates for the Calabi–Yau metrics due to Kobayashi. As an application, we study stable closed geodesics. We use the metric estimates to show how there are generally restrictions on the existence of such geodesics. We also show how there can exist stable, closed geodesics in some highly symmetric circumstances due to hyperkähler identities.

In this article we introduce conformal Bach flow and establish its well-posedness on closed manifolds. We also obtain its backward uniqueness. To give an attempt to study the long-time behavior of conformal Bach flow, assuming that the curvature and the pressure function are bounded, global and local Shi’s type (L^2)-estimate of derivatives of curvatures is derived. Furthermore, using the (L^2)-estimate and based on an idea from (Streets in Calc Var PDE 46:39–54, 2013) we show Shi’s pointwise estimate of derivatives of curvatures without assuming Sobolev constant bound.