Solvability of Minimal Graph Equation Under Pointwise Pinching Condition for Sectional Curvatures

Abstract

Abstract. We study the asymptotic Dirichlet problem for the minimal graph equation on a Cartan-Hadamard manifold M whose radial sectional curvatures outside a compact set satisfy an upper boundand a pointwise pinching conditionfor some constants φ > 1 and C K ≥ 1, where P and P are any 2-dimensional subspaces of TxM containing the (radial) vector ∇r(x) and r(x) = d(o, x) is the distance to a fixed point o ∈ M . We solve the asymptotic Dirichlet problem with any continuous boundary data for dimensions n = dim M > 4/φ + 1.

Bernstein and half-space properties for minimal graphs under Ricci lower bounds

“…The interested reader may consult [7,Thm. 1.5] and [7,9], as well as [27] for related interesting non-existence results. This naturally motivates the following Question.…”

Existence and non-existence of minimal graphic and p-harmonic functions

“…Recently the asymptotic Dirichlet problem for minimal graph, f -minimal graph, p-harmonic and A-harmonic equations has been studied for example in [22], [30], [31], [7], [21], [8], [6], [5], and [17], where the existence of solutions was studied under various curvature assumptions and via different methods. In [8] the existence of solutions to the minimal graph equation and to the A-harmonic equation was proved in dimensions n ≥ 3 under curvature assumptions − log r(x)) 2ε r(x) 2 ≤ K(P x ) ≤ − 1 + ε r(x) 2 log r(x)…”

Asymptotic Dirichlet Problem for 𝓐 $\mathcal {A}$ -Harmonic Functions on Manifolds with Pinched Curvature

“…Cheng [5] was the first to solve the problem for the Laplacian under the same type of pointwise pinching assumption for the sectional curvatures as we consider in this paper. Later the asymptotic Dirichlet problem has been generalized for p-harmonic and A-harmonic functions and for minimal graph equation under various curvature assumptions, see [2], [3], [10], [11], [14], [15].…”

“…where φ > 1 is constant. In [2] the authors showed that, with these weaker assumptions, the solvability result holds also for the minimal graph equation.…”

“…In this paper we will use similar techniques as in [2], [3] and [15]. Our main theorem is the following.…”