Einstein-Type Structures, Besse’s Conjecture, and a Uniqueness Result for a $$\varphi $$-CPE Metric in Its Conformal Class

Colombo, Giulio; Mari, Luciano; Rigoli, Marco

doi:10.1007/s12220-022-01000-3

Cited by 3 publications

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“…In the recent paper [4] (Lemma 8), the following theorem was established. Let M be a complete Riemannian manifold (without boundary), λ > 0 a constant and u ∈ C 2 (M).…”

Section: Introductionmentioning

confidence: 99%

“…This can be regarded as a sort of "gap" theorem for subsolutions of u = λu: if u ∈ C 2 (M) satisfies u ≥ λu on M then either u ≤ 0 or the positive part of u has to be sufficiently large in an integral sense (that is, its L 2 norm on B R (x 0 ) must grow at least exponentially with respect to R). In fact, the result from [4] is more general and also covers the case of weighted Laplacians and locally Lipschitz weak solutions of (1).…”

Section: Introductionmentioning

confidence: 99%

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Growth of Subsolutions of $$\Delta _p u = V|u|^{p-2}u$$ and of a General Class of Quasilinear Equations

Alías,

Colombo,

Rigoli

2023

J Geom Anal

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In this paper we prove some integral estimates on the minimal growth of the positive part $$u_+$$ u + of subsolutions of quasilinear equations $$\begin{aligned} \textrm{div}A(x,u,\nabla u) = V|u|^{p-2}u \end{aligned}$$ div A ( x , u , ∇ u ) = V | u | p - 2 u on complete Riemannian manifolds M, in the non-trivial case $$u_+\not \equiv 0$$ u + ≢ 0 . Here A satisfies the structural assumption $$|A(x,u,\nabla u)|^{p/(p-1)} \le k \langle A(x,u,\nabla u),\nabla u\rangle $$ | A ( x , u , ∇ u ) | p / ( p - 1 ) ≤ k ⟨ A ( x , u , ∇ u ) , ∇ u ⟩ for some constant $$k>0$$ k > 0 and for $$p>1$$ p > 1 the same exponent appearing on the RHS of the equation, and V is a continuous positive function, possibly decaying at a controlled rate at infinity. We underline that the equation may be degenerate and that our arguments do not require any geometric assumption on M beyond completeness of the metric. From these results we also deduce a Liouville-type theorem for sufficiently slowly growing solutions.

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“…In the recent paper [4] (Lemma 8), the following theorem was established. Let M be a complete Riemannian manifold (without boundary), λ > 0 a constant and u ∈ C 2 (M).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%