What question did this study set out to answer?

To establish relationships between harmonic functions defined on two subsurfaces of a Riemann surface using Weil–Petersson curves.

March 3, 2026Open Access

Weil–Petersson Curves and Dirichlet Finite Harmonic Functions on Riemann Surfaces

Puntos clave

To establish relationships between harmonic functions defined on two subsurfaces of a Riemann surface using Weil–Petersson curves.
Analyzed harmonic functions in Banach spaces on the subsurfaces divided by a p-Weil–Petersson curve.
Considered the p-Dirichlet integrals' finiteness in the complementary domains of the curve.
Investigated the composition operator acting on Banach space of p-Besov functions.
Established correspondence between harmonic functions based on boundary conditions on the Weil–Petersson curve.
Analyzed the implications of the correspondence through functional analysis of operators.
Provided insights into the behavior of harmonic functions on complex surfaces.

Resumen

On two subsurfaces of a Riemann surface divided by a p-Weil–Petersson curve γ, we consider the spaces of harmonic functions whose p-Dirichlet integrals are finite in the complementary domains of γ. By requiring the coincidence of boundary values on γ, we establish a correspondence between the harmonic functions in these Banach spaces. We analyze the operator arising from this correspondence via the composition operator acting on the Banach space of p-Besov functions on the unit circle.

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