Manifolds with Density

Like others interested in this topic, I first learned about Manifolds with Density from Dr. Frank Morgan — in my case this happened when I was a graduate student at Rice University.  The idea of re-examining Riemannian Geometry from a new point of view appealed to me, and I regularly found myself learning more from Dr. Morgan and his SMALL students about this idea over the ensuing years.

An important example of this subject concerns one-dimensional curves that live inside of a plane, \mathbb{R}^2.  However, this is no ordinary plane.  One first equips \mathbb{R}^2 with a density function, \Phi:\mathbb{R}^2 \to \mathbb{R}^+, and uses it to weight lengths and areas in a curious manner.  Given a curve \gamma \subset \mathbb{R}^2 and a region R \subseteq \mathbb{R}^2 the respective \Phi-weighted length and area are given by

\displaystyle \text{length}(\gamma) = \int_{\gamma} \! \Phi \,ds

\displaystyle \text{area}(R ) = \iint_R \! \Phi \, dA

As mentioned in the linked article above, this is a curious choice to make since, usually, density functions \Phi scale in accordance with the dimension of the subset being measured.  In the second integral, for instance, we would expect \Phi^2 to appear.  Indeed, it is precisely this decision that makes life so interesting in these spaces. One can (and many have) study \Phi-weighted isoperimetric problems in this strange plane, and many of the standard techniques still apply.  For instance, using a first-variation argument, one can show that any curve that solves an \Phi-weigthed isoperimetric problem must have constant \Phi-weighted curvature.

When \displaystyle \Phi(x,y) = e^{x^2+y^2} is a Gaussian density, the equation for \Phi-weighted curvature turns into a shockingly beautiful one.  Such a curve satisfies a modified self-shrinker equation.  In short, this means that curves with Gaussian-weighted-curvature identically equal to zero have been studied for quite some time, they are precisely those curves that shrink under mean curvature flow.  After learning more about these ideas while working with students at St. Mary’s, I wrote a small paper on constant (non-zero) Gaussian-weighted curvature curves and hypersurfaces in Gaussian space (a copy of which can be found here).  The image atop this webpage shows a constant Gaussian-weighted curvature cylinder that is obtained by stacking up an infinite collection of constant Gaussian-weighted curves and was taken from this paper

I have also submitted a more general-audience-friendly paper on another way to approach such objects.  These curves can be viewed as interesting generalizations of a circle (centered at the origin) in the standard Euclidean plane.  Like a circle, these curves sweep out polar area in proportion to their turning, and, as such, I refer to them as “turning sweepers” (not especially creative, I know).  In addition to deploying undergraduate-friendly techniques to understand such curves, I also study curves that sweep area with respect to multiple points (not simply the origin) in proportion with their turning.

Of course, there is no reason to limit this view to curves in a plane.  In fact, I have worked with multiple students on different ways to generalize these ideas to space curves as well as curves in general manifolds.  For example, with math major Dylan Weber, I examined space curves ( \gamma \subset \mathbb{R}^3 ) that sweep out surface area (with respect to, say, the origin) in proportion to other geometric quantities.  We expect this work to give rise to new creatures, so-called “twisting sweepers,” and hope to understand how they, too, may or may not be related to manifolds with density.

— Casey