Schauder estimates on products of cones

Abstract

We prove an interior Schauder estimate for the Laplacian on metric products of two dimensional cones with a Euclidean factor, generalizing the work of Donaldson and reproving the Schauder estimate of Guo-Song. We characterize the space of homogeneous subquadratic harmonic functions on products of cones, and identify scales at which geodesic balls can be well approximated by balls centered at the apex of an appropriate model cone. We then locally approximate solutions by subquadratic harmonic functions at these scales to measure the Hölder continuity of second derivatives.

Calabi‐Yau metrics with cone singularities along intersecting complex lines: The unstable case

“…In this section we establish our main estimate, Proposition 4.11. We prove it by a perturbation method, along the lines of [12], which has the advantage of being robust and (contrary to [15])…”

“…We begin by recalling the space of subquadratic homogeneous harmonic functions on 𝐂 𝛽 × 𝐂 𝛾 with angles 0 < 𝛽 ⩽ 1 and 0 < 𝛾 ⩽ 1 as in [12,Section 3]. † We regard 𝐂 𝛽 × 𝐂 𝛾 as a cone whose link † Here we discuss the general picture, but later we will restrict to 𝛽 = 𝛽 1 , 𝛽 2 , 𝛽 3 , 1 and 𝛾 = 𝛾, 1 depending on the model cone.…”

“…The above proposition is proved by separation of variables as we sketch below, see [12,Proposition 3.4] for more details. A key property that we will exploit is that |𝜕 ∂𝑃| 𝛼 = 0 for every 𝑃 ∈  ⩽2 .…”

“…We begin by recalling the space of subquadratic homogeneous harmonic functions on $$\mathbf {C}_{\beta } \times \mathbf {C}_{\gamma }$$ with angles $$0&lt;\beta \leqslant 1$$ and $$0&lt;\gamma \leqslant 1$$ as in [12, Section 3]. We regard $$\mathbf {C}_{\beta } \times \mathbf {C}_{\gamma }$$ as a cone whose link is a three‐sphere endowed with a constant sectional curvature $$\hskip.001pt 1$$ metric with conical singularities along two Hopf circles (or only one if either $$\gamma =1$$ or $$\beta =1$$, and none if $$\beta =\gamma =1$$).…”

“…The main technical work of the paper is in Section 4 deriving an appropriate Schauder estimate for the Laplace operator of the approximate solution acting on Hölder spaces. The estimate is proved by Campanato iteration using harmonic approximations, along the lines of [12]. There are two key ingredients: (i) approximation of balls, up to a fixed error, by balls centred at the apex of suitable model cones; (ii) $$C^{\alpha }$$ control on the complex Hessian for a family of reference functions which approximate the subquadratic harmonic polynomials on the model cones.…”