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Networks in Manifolds with Density

14th January 2009, 09:13 am

Abstract: The version of the shortest “Steiner” network problem in which you minimize length plus number of Steiner points has an interesting analog in manifolds with density.On a Riemannian manifold M, a density f is fundamentally different from a conformal change of metric \lambda because for every m, m-dimensional area is weighted by simply f in contrast to different powers \lambda^m of \lambda. This difference shows up in any problem that involves objects of different dimensions, such as the isoperimetric problem, which involves both volume and perimeter. Similarly, it shows up in the network problem of minimizing (weighted) length plus the sum of the weights of additional Steiner points. The equilibrium condition at a Steiner point with edges going in unit directions v_i is that

     \Sigma v_i + \nabla f = 0.

This problem arose at a lunch at the 2009 annual joint mathematics meeetings in DC with Konrad Swanepoel, Max Engelstein, Scott Greenleaf, Neil Hoffman, and Bret Thacher.

Category: Math

2 Comments

  1. Sam Ferguson:

    Frank,

    Is there some way to combine Lagrange multipliers with the Calculus of Variations that would allow us to have a whack at (particular instances of) this problem?

    Also, I found an REU on “Steiner Problems,” to be held at Brigham Young University this summer, and it might be worthwhile to apply.

    Is there an REU on such problems at Williams?

    3 February 2009, 11:08 pm

  2. Frank Morgan:

    Hi Sam. Yes, Lagrange multipliers is one way to get the equilibrium condition. The BYU REU is a good one. For mine at Williams (actually in Granada Spain this summer), choose the “SMALL” link on our department webpage.

    FM

    4 February 2009, 10:23 am