🔢Potential Theory Unit 9 – Potential Theory on Riemannian Manifolds

Key Concepts and Definitions

• Riemannian manifold: A smooth manifold equipped with a Riemannian metric, which is a positive definite inner product on each tangent space
• Potential function: A scalar function defined on a manifold whose gradient gives a conservative vector field
• Laplace-Beltrami operator: The generalization of the Laplace operator to Riemannian manifolds, defined as the divergence of the gradient
  • Plays a crucial role in the study of harmonic functions and the Dirichlet problem
• Harmonic function: A function that satisfies the Laplace-Beltrami equation, i.e., its Laplace-Beltrami operator is zero
• Green's function: A fundamental solution to the Laplace-Beltrami operator, used to solve inhomogeneous differential equations
• Dirichlet problem: The problem of finding a harmonic function on a domain with prescribed values on the boundary
• Geodesic: The shortest path between two points on a Riemannian manifold, generalizing the concept of a straight line in Euclidean space

Riemannian Manifold Basics

• Tangent space: A vector space attached to each point of a manifold, consisting of all possible velocity vectors of curves passing through that point
• Riemannian metric: A smoothly varying inner product on the tangent spaces, allowing the measurement of lengths, angles, and volumes on the manifold
  • Induces a distance function on the manifold, making it a metric space
• Levi-Civita connection: A unique connection on a Riemannian manifold that is compatible with the metric and torsion-free
• Curvature: A measure of how the manifold deviates from being Euclidean, quantified by the Riemann curvature tensor
• Geodesic equation: A second-order differential equation describing the paths of geodesics on the manifold
• Exponential map: A map from the tangent space to the manifold, sending a tangent vector to the endpoint of the geodesic starting at the base point with the given initial velocity
• Parallel transport: A way to move tangent vectors along curves on the manifold while preserving their lengths and angles, using the Levi-Civita connection

Potential Functions on Manifolds

• Conservative vector field: A vector field that is the gradient of some scalar potential function
  • The work done by a conservative vector field along any closed path is zero
• Gradient: The generalization of the usual gradient to Riemannian manifolds, defined using the Riemannian metric
• Laplacian: The Laplace-Beltrami operator applied to a potential function, measuring the divergence of its gradient field
• Poisson equation: An inhomogeneous differential equation involving the Laplace-Beltrami operator, $\Delta u = f$, where $f$ is a given function on the manifold
• Newtonian potential: The gravitational potential function in Newtonian mechanics, satisfying the Poisson equation with the mass density as the source term
• Electrostatic potential: The electric potential function in electrostatics, satisfying the Poisson equation with the charge density as the source term
• Dirichlet energy: The energy functional associated with a potential function, given by the $L^2$ norm of its gradient

Laplace-Beltrami Operator

• Divergence: The generalization of the usual divergence to Riemannian manifolds, measuring the net outward flux of a vector field per unit volume
• Laplace-Beltrami equation: The homogeneous differential equation $\Delta u = 0$, characterizing harmonic functions on the manifold
• Eigenvalues and eigenfunctions: The Laplace-Beltrami operator has a discrete spectrum of eigenvalues and corresponding eigenfunctions on compact manifolds
  • Eigenfunctions form a complete orthonormal basis for the space of square-integrable functions on the manifold
• Heat equation: A parabolic partial differential equation involving the Laplace-Beltrami operator, describing the evolution of temperature on the manifold
• Wave equation: A hyperbolic partial differential equation involving the Laplace-Beltrami operator, describing the propagation of waves on the manifold
• Schrodinger equation: A partial differential equation involving the Laplace-Beltrami operator, describing the quantum mechanics of a particle on the manifold
• Bochner's formula: An identity relating the Laplacian of the norm of a vector field to its divergence and the Ricci curvature of the manifold

Harmonic Functions and Green's Functions

• Maximum principle: A harmonic function attains its maximum and minimum values on the boundary of its domain
  • Implies that a harmonic function is uniquely determined by its boundary values
• Mean value property: The value of a harmonic function at a point is equal to its average value over any sphere centered at that point
• Harnack's inequality: An estimate comparing the values of a positive harmonic function at two different points in terms of the distance between them
• Poisson kernel: An integral kernel used to express the solution to the Dirichlet problem in terms of the boundary values
• Fundamental solution: A Green's function with a singularity at a given point, satisfying $\Delta G(x,y) = \delta(x-y)$ where $\delta$ is the Dirac delta function
• Green's representation formula: Expresses a function in terms of its boundary values and the Green's function of the domain
• Green's identities: Relate the Laplacian of a function to its boundary values and normal derivatives, using integration by parts on the manifold

Dirichlet Problem on Manifolds

• Existence and uniqueness: The Dirichlet problem has a unique solution for continuous boundary data on a compact manifold with boundary
  • Follows from the maximum principle and the solvability of the Poisson equation
• Perron's method: A constructive approach to solving the Dirichlet problem, based on finding the supremum of all subharmonic functions lying below the boundary data
• Capacity: A measure of the size of a subset of the boundary, related to the solvability of the Dirichlet problem with boundary data concentrated on that subset
• Regularity: The solution to the Dirichlet problem inherits the regularity of the boundary data and the manifold
  • Smooth boundary data and manifold imply a smooth solution
• Variational formulation: The Dirichlet problem can be posed as a variational problem, minimizing the Dirichlet energy among all functions with the given boundary values
• Finite element method: A numerical approach to solving the Dirichlet problem, based on discretizing the manifold into a mesh and approximating the solution by piecewise polynomial functions
• Probabilistic interpretation: The solution to the Dirichlet problem can be interpreted as the expected value of a random walk on the manifold, starting from a given point and stopping upon reaching the boundary

Applications in Physics and Geometry

• Electrostatics: The electrostatic potential of a charge distribution on a manifold satisfies the Poisson equation, with the Dirichlet problem corresponding to grounded conductors
• Gravitation: The Newtonian gravitational potential of a mass distribution on a manifold satisfies the Poisson equation, with the Dirichlet problem corresponding to fixed potential surfaces
• Quantum mechanics: The wavefunction of a quantum particle on a manifold satisfies the Schrodinger equation, with the Dirichlet problem corresponding to fixed boundary conditions
• Heat conduction: The temperature distribution on a manifold evolves according to the heat equation, with the Dirichlet problem corresponding to fixed temperature boundaries
• Minimal surfaces: A minimal surface is a surface with zero mean curvature, which can be characterized as a harmonic map from a domain to the manifold
  • The Dirichlet problem for harmonic maps corresponds to finding minimal surfaces with fixed boundary curves
• Ricci flow: A geometric evolution equation for Riemannian metrics, with the Laplace-Beltrami operator playing a key role in the evolution of curvature
• Spectral geometry: The study of the relationship between the spectrum of the Laplace-Beltrami operator and the geometry of the manifold, with applications to shape analysis and inverse problems

Advanced Topics and Open Problems

• Nonlinear potential theory: The study of potential functions satisfying nonlinear equations, such as the $p$-Laplace equation or the mean curvature equation
• Singular perturbation theory: The study of the behavior of solutions to the Poisson equation or the Dirichlet problem as the manifold undergoes a singular limit, such as a collapsing or degenerating sequence of metrics
• Stochastic PDEs: The study of partial differential equations with random coefficients or noise terms, such as the stochastic heat equation or the stochastic Poisson equation on manifolds
• Fractional Laplacians: Generalizations of the Laplace-Beltrami operator to non-integer powers, leading to the study of fractional diffusion and fractional quantum mechanics on manifolds
• Infinite-dimensional manifolds: The study of potential theory on infinite-dimensional manifolds, such as the space of Riemannian metrics or the space of shapes, with applications to geometric analysis and shape optimization
• Dirichlet-to-Neumann maps: The study of the relationship between the Dirichlet and Neumann boundary value problems, with applications to inverse problems and electrical impedance tomography
• Solvability of the Dirichlet problem for rough boundaries or metrics: The question of whether the Dirichlet problem is solvable for irregular boundary data or singular Riemannian metrics, leading to the development of new function spaces and regularity theories