Math 2.0

Potential theory Definition

Harmonic functions on Riemannian manifolds

Builds on Riemannian manifoldLaplace–Beltrami operatorHarmonic function

Definition

Let (M,g)(M, g) be a Riemannian manifold and let U⊆MU \subseteq M be an open set. In local coordinates x1,…,xnx^1, \dots, x^n, write gijg_{ij} for the metric, gijg^{ij} for its inverse matrix and ∣g∣=det⁡(gij)\abs{g} = \det(g_{ij}). The Laplace–Beltrami operator of a function u∈C2(U)u \in C^2(U) is Δgu=1∣g∣ ∂i(∣g∣ gij ∂ju),\Delta_g u = \frac{1}{\sqrt{\abs{g}}}\, \partial_i \Big( \sqrt{\abs{g}}\, g^{ij}\, \partial_j u \Big), with summation over repeated indices; it equals div⁡ggrad⁡gu\operatorname{div}_g \operatorname{grad}_g u and does not depend on the coordinates. A function u∈C2(U)u \in C^2(U) is harmonic on UU if Δgu=0\Delta_g u = 0 on UU.

Remarks

For M=ℝnM = \R^n with the Euclidean metric, Δg\Delta_g is the usual Laplacian, and the definition gives the classical harmonic functions. Example: on the upper half-plane {(x,y):y>0}\set{(x, y) : y > 0} with the hyperbolic metric g=y−2(dx2+dy2)g = y^{-2}(dx^2 + dy^2), one has Δg=y2(∂x2+∂y2)\Delta_g = y^2(\partial_x^2 + \partial_y^2). So the function u(x,y)=xu(x, y) = x is harmonic, and so is the bounded function arctan⁡(x/y)\arctan(x/y). Non-example: on the unit sphere S2⊂ℝ3S^2 \subset \R^3 with the round metric, the height function zz satisfies Δgz=−2z\Delta_g z = -2z, so it is not harmonic. On a connected closed manifold every harmonic function is constant, since ∫M∣∇u∣2 dVg=−∫Mu Δgu dVg=0\int_M \abs{\nabla u}^2 \, dV_g = -\int_M u\, \Delta_g u \, dV_g = 0. The definition depends on the metric: a function harmonic for one metric need not be harmonic for another. In dimension 22, however, harmonicity is unchanged under conformal changes of metric.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptLaplace–Beltrami operator, concept, no evidence levelLaplace–Beltrami oper…conceptRiemannian manifold, concept, no evidence levelRiemannian manifoldconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptLaplacian, concept, no evidence levelLaplacianconceptHarmonic function, concept, no evidence levelHarmonic functionconceptHarmonic functions on Riemannian manifolds, definition, no evidence levelHarmonic functions on…definition
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  1. Euclidean space object
  2. Function concept
  3. Laplace–Beltrami operator concept
  4. Riemannian manifold concept
  5. Total derivative concept
  6. Function of several variables concept
  7. Vector-valued function concept
  8. Differentiable function of several variables concept
  9. Partial derivative concept
  10. Vector field concept
  11. Gradient concept
  12. Divergence concept
  13. Laplacian concept
  14. Harmonic function concept
  15. Harmonic functions on Riemannian manifolds definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Laplace%E2%80%93Beltrami_operator
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MSC31C12, 58J05
DefinesHarmonic functions on Riemannian manifolds
UsesRiemannian manifold, Laplace–Beltrami operator, Harmonic function

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