Conformal metrics with prescribed boundary mean curvature on balls

Abstract

We provide a variety of classes of functions that can be realized as the mean curvature on the boundary of the standard n dimensional ball, n ≥ 3, with respect to some scalar flat metric. Because of the presence of some critical nonlinearity, blow up phenomena occur and existence results are highly nontrivial since one has to overcome topological obstructions. Our approach consists of, on one hand, developing a Morse theoretical approach to this problem through a Morse-type reduction of the associated Euler-Lagrange functional in a neighborhood of its critical points at Infinity and, on the other hand, extending to this problem some topological invariants introduced by A. Bahri in his study of Yamabe type problems on closed manifolds.Keywords Boundary mean curvature · Intersection number · Morse theory · Morse lemma at infinity · Critical points at infinity · Morse inequalities · Boundary operator · Topological methods Mathematics Subject Classification (2000) 58E05 · 35J65 · 53C21 · 35B40

A Morse theoretical approach for the boundary mean curvature problem on B4

“…Since J has no critical points in Σ + , it follows from [14,Proposition 7.24 and Theorem 8.2], that J c 1 = {u ∈ Σ + | J (u) c 1 } retract by deformation onto X ∞ = W u (y 1 ) ∞ ∪ W u (y 0 ) ∞ , which can be parameterized by X × [A, +∞[. Now, we claim that X ∞ is contractible in J c 1 Since H (x) H (y 1 ) for any x ∈ X, it follows from the above estimates that J (f (t, x, λ)) < c 1 for any (t, x, λ) ∈ [0, 1] × X × [A, ∞[.…”

“…In this paper, as well as in its second part [1], we are interested in the case of standard balls where the noncompact group of conformal transformations of the ball, acts on the equation giving rise to Kazdan-Warner type obstructions, just as in the celebrated scalar curvature (or Nirenberg) problem (see [39]). Namely, let B n be the unit ball in R n with Euclidean metric g 0 .…”

ON THE PRESCRIBED MEAN CURVATURE PROBLEM ON THE STANDARD n-DIMENSIONAL BALL

“…The behavior of sequences failing the Palais-Smale condition can be characterized taking into account the uniqueness result of Li and Zhu [13] and following the ideas introduced in [2]. We have the following proposition:…”

PRESCRIBING MEAN CURVATURE ON 𝔹n

“…In [21], Escobar has studied this problem (1.2) on manifolds which are not equivalent to the standard ball. On the ball, one group of existence results have been obtained under the following non-degeneracy condition: ∆H(y) = 0 whenever ∇H(y) = 0, (nd) see [3,22] for n = 3, [2,18] for n = 4 and [1,13] for n ≥ 3. For example in [3], it is assumed that H is a C 2 Morse function with the (nd) condition.…”

“…On the ball, one group of existence results have been obtained under the following non-degeneracy condition: ∆H(y) = 0 whenever ∇H(y) = 0, (nd) see [3,22] for n = 3, [2,18] for n = 4 and [1,13] for n ≥ 3. For example in [3], it is assumed that H is a C 2 Morse function with the (nd) condition. Then, if ind(H, y) denotes the Morse index of H at the critical point y, the problem (1.2) on the standard ball B 3 has a solution provided that ∆H(y)<0…”