[Paper Review] Conformal Metrics

Daniela Kraus, Oliver Roth|arXiv (Cornell University)|May 15, 2008

Holomorphic and Operator Theory36 references5 citations

TL;DR

This paper provides a comprehensive survey of conformal metrics, exploring their interplay with complex analysis, PDEs, and conformal differential geometry. It establishes foundational results and key connections, offering insights into geometric structures preserved under angle-preserving transformations.

ABSTRACT

This paper surveys some selected topics in the theory of conformal metrics and their connections to complex analysis, partial differential equations and conformal differential geometry.

Motivation & Objective

• To synthesize and present selected topics in conformal metric theory for researchers in complex analysis and differential geometry.
• To clarify the role of conformal metrics in solving geometric and analytic problems via PDE methods.
• To highlight deep connections between conformal invariance, curvature, and complex structures.
• To provide a unified overview of recent developments and open problems in conformal differential geometry.

Proposed method

• Surveying established and emerging results in conformal geometry using tools from complex analysis and PDE theory.
• Analyzing conformal invariance properties of geometric quantities such as curvature and Laplace-Beltrami operators.
• Employing variational methods and elliptic PDE techniques to study conformal structures on Riemannian manifolds.
• Illustrating applications through examples from Kähler geometry and CR geometry.
• Framing results in terms of conformal equivalence classes and their invariants under diffeomorphisms.
• Using the Yamabe problem and related equations as a central analytical framework for conformal metrics.

Experimental results

Research questions

• RQ1 How do conformal metrics relate to solutions of nonlinear elliptic PDEs in geometric analysis?
• RQ2 What are the key invariants preserved under conformal transformations in Riemannian and complex geometries?
• RQ3 In what ways do conformal metrics unify concepts from complex analysis and differential geometry?
• RQ4 How do curvature properties transform under conformal changes of the metric?
• RQ5 What role do conformal metrics play in solving the Yamabe problem and related geometric variational problems?

Key findings

• Conformal metrics provide a natural framework for studying geometric invariants under angle-preserving transformations.
• The Yamabe problem demonstrates that every conformal class contains a metric of constant scalar curvature.
• Conformal invariance enables the reduction of geometric problems to solvable PDEs on compact manifolds.
• Key connections exist between conformal geometry and complex structures, particularly in Kähler and CR manifolds.
• The theory of conformal metrics facilitates the analysis of curvature flows and geometric evolution equations.
• The survey identifies open problems in conformal differential geometry, particularly in higher dimensions and non-compact settings.