Splitting lemmas for the Finsler energy functional on the space ofH¹-curves

Abstract

We establish the splitting lemmas (or generalized Morse lemmas) for the energy functionals of Finsler metrics on the natural Hilbert manifolds of H1‐curves around a critical point or a critical double-struckR1 orbit of a Finsler isometry‐invariant closed geodesic. They are the desired generalization on Finsler manifolds of the corresponding Gromoll–Meyer's splitting lemmas on Riemannian manifolds [Gromoll and Meyer, ‘On differentiable functions with isolated critical points’, Topology 8 (1969) 361–369; Gromoll and Meyer ‘Periodic geodesics on compact Riemannian manifolds’, J. Differential Geom. 3 (1969) 493–510]. As an application, we extend to Finsler manifolds a result by Grove and Tanaka [‘On the number of invariant closed geodesics’, Acta Math. 140 (1978) 33–48; Tanaka, ‘On the existence of infinitely many isometry‐invariant geodesics’, J. Differential Geom. 17 (1982) 171–184] about the existence of infinitely many, geometrically distinct, isometry invariant closed geodesics on a closed Riemannian manifold.

Parameterized splitting theorems and bifurcations for potential operators, Part I: Abstract theory

“…In applications to Lagrange systems using Theorem 4.6 and Corollaries 4.7, 4.8 produce stronger results than using Theorem 4.2 and Corollaries 4.3, 4.4 (see [47]). Moreover, we may give parameterized versions of splitting lemmas for the Finsler energy functional in [44] as Theorem A.3 and then obtain similar results to Theorem 4.6 and Corollaries 4.7, 4.8 for geodesics on Finsler manifolds. Now we give a multi-parameter bifurcation result, which is not only a converse of Corollary 3.4 in stronger conditions, but also a generalizations of Corollaries 4.7, 4.8 in some sense.…”

Parameterized splitting theorems and bifurcations for potential operators

“…As in Theorem 2.3, many bifurcation theorems in last three sections can be given in the setting of [37,38], which is more suitable for variational problems in Finsler geometry ( [41]). We are only satisfied to state a few of them.…”

Morse theory methods for a class of quasi-linear elliptic systems of higher order

“…Comparing with splitting lemmas in [32,33], the new ones may largely simplify the arguments for Lagrangian systems in [32]. However, the former may, sometime, provide more elaborate results, for example, as we have done modifying the proof ideas of them may yield the desired splitting lemma for the Finsler energy functional on the space of H 1 -curves in [36]. It is not clear how to complete this with the present one.…”

“…Even if for the Lagrangian systems studied in [39], we can largely simplify the arguments therein with this new theorem. However, the theories in [39,40] may, sometime, provide more elaborate results as done in [44,46,47,48].…”