We study 1-parameter families in the space M G 1 of G-invariant, unit volume metrics on a given compact, connected, almost-effective homogeneous space M = G/H. In particular, we focus on diverging sequences, i.e. that are not contained in any compact subset of M G 1 , and we prove some structure results for those which have bounded curvature. We also relate our results to an algebraic version of collapse. 1 ) ε := {g ∈ M G 1 : scal(g) ≥ ε}, with ε > 0. Namely, if (g (n) ) ⊂ M G 1 is a sequence for which scal(g (n) ) → ε and Ric o (g (n) ) g (n) → 0, then one can extract a subsequence which converges in the C ∞ -topology to an Einstein metric g (∞) ∈ M G 1 with scal(g (∞) ) = ε > 0 [7, Thm A]. Here, Ric o (g (n) ) is the traceless Ricci tensor of g (n) and | · | g (n) is the norm induced by g (n) on the tensor bundle over M . As is well known, the traceless Ricci tensor is precisely the negative gradient vector of the functional scal with respect to the standard L 2 -metric · , · .On the other hand, again in [7], the authors also studied the so called 0-Palais-Smale sequences, i.e. (g (n) ) ⊂ M G 1 such that scal(g (n) ) → 0 and Ric o (g (n) ) g (n) → 0. Notice that, unlike the previous case, a 0-Palais-Smale sequence (g (n) ) cannot have convergent subsequences if M is not a torus. This means that (g (n) ) goes off to infinity on the set M G 1 and consequently we say that such sequences are divergent. Remarkably, there are topological obstructions on the existence of 0-Palais-Smale sequences. In fact by [7, Thm 2.1] if M admits a 0-Palais-Smale sequence, then there exists a closed, connected intermediate subgroupa torus. Here, H o and G o denote the identity components of H and G, respectively. This last theorem is optimal if the isotropy group H is connected. In case H is disconnected, the authors conjectured that G/H is itself a homogeneous torus bundle [7, p. 697]. The first main result proved in this paper for the purpose of generalizing [7, Thm 2.1] is Theorem A. Let M m = G/H be a compact, connected homogenous space. If there exists a diverging sequence (g (n) ) ⊂ M G 1 with bounded curvature, i.e. with | sec(g (n) )| ≤ C for some constant C > 0, then there exists an intermediate closed subgroup H K ⊂ G such that the quotient K/H is a torus.We stress that the proof of Theorem A is purely algebraic and constructive. In fact, we show that the sum of the eigenspaces associated to all the shrinking eigenvalues of any diverging sequence (g (n) ) ⊂ M G 1 with bounded curvature is a reductive complement of h = Lie(H) into an intermediate Ad(H)-invariant Lie 2010 Mathematics Subject Classification. 53C30, 53C21, 57S15. Key words and phrases. Compact homogenous spaces, invariant Riemannian metrics, curvature bounds. This work was supported by GNSAGA of INdAM. 1 DIVERGING SEQUENCES OF UNIT VOLUME INVARIANT METRICS WITH BOUNDED CURVATURE 3 g (n) -Killing vector fields X (n) induced by the action of G on M such thatRoughly speaking it means that, up to normalize with respect to the 1-jet norm, the sequence (X (n) ) is ...
We prove the Myers-Steenrod theorem for local topological groups of isometries acting on pointed C k,α-Riemannian manifolds, with k + α > 0. As an application, we infer a new regularity result for a certain class of locally homogeneous Riemannian metrics.
In this note, we begin the study of Hermitian manifolds with a compact Lie group action by holomorphic isometries with principal orbits of codimension one. In particular, we focus on a special class of these manifolds constructed by following Bérard-Bergery: we compute the Chern-Ricci curvatures and we characterize the special Hermitian metrics, such as balanced, Kähler-like, pluriclosed, locally conformally Kähler, Vaisman, Gauduchon.
In this paper, we study Sobolev regularity of solutions to nonlinear second order elliptic equations with super-linear first-order terms on Riemannian manifolds, complemented with Neumann boundary conditions, when the source term of the equation belongs to a Lebesgue scale, under various integrability regimes. Our method is based on an integral refinement of the Bernstein method, and leads to "semilinear Calderón-Zygmund" type results. Applications to the problem of smoothness of solutions to Mean Field Games systems with Neumann boundary conditions posed on convex domains of the Euclidean space will also be discussed.
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