Research Statement
Brian Harvie
My research interests lie in the field of geometric analysis, specifically the study of geometric flows and their applications in general relativity. My dissertation work has focused on a partic- ular geometric flow called Inverse Mean Curvature Flow, and so I will begin with a background on this flow while also introducing Mean Curvature Flow to serve as a point of comparison.
1 Introduction: Mean Curvature Flow vs Inverse Mean Curva- ture Flow
A geometric flow is a rule for deforming a surface by its curvature over time. Geometric flows arise in two flavors: intrinsic and extrinsic. My research focus is on extrinsic geometric flows, particularly over an n-dimensional closed hypersurface M0 ⊂ Rn+1. The way M0 deforms through time under an extrinsic flow depends on its extrinsic curvature, a measure of how the surface bends with respect to the ambient space at each point.
Extrinsic curvature at each p ∈ M0 is encoded in n distinct principal curvatures λi(p) of M0
at p. These are defined as the eigenvalues of a tensor over M0known as the Second Fundamental Form A, but there is an especially helpful geometric interpretation of the principal curvatures for a surface in R3: at each point on the surface, λ1 and λ2 measure how the normal vector ν of the surface bends at that point as one moves in two orthogonal directions on M0, see Figure 1.
Definition 1.1. The mean curvature H : M0 → R is the function H(p) = Σni=1λi(p).
So, up to a factor of n1, H is the arithmetic mean of all the principal curvatures at p ∈ M0. It is natural to consider the evolution of a surface M0 through time under a geometric flow where the flow speed at each point p ∈ M0 is some function of the mean curvature H at p. One such example, called Mean Curvature Flow, will cause the surface to contract through time, while another closely-related flow, called Inverse Mean Curvature Flow, will cause the surface to expand.
Figure 1: The principal curvatures λ1 and λ2 at a point p ∈ M equal the curvatures for two curves intersecting orthogonally at p. Image from [Vas].
Definition 1.2. Given a closed, oriented n-dimensional smooth manifold M , a one-parameter family of immersions F : M ×[0, T ) → Rn+1is a solution to Mean Curvature Flow (MCF) if
∂Ft
∂t (p, t) = −Hν(p, t), (p, t) ∈ M × [0, T ), (1) where ν is the outward unit normal of Mt= Ft(M ).
Definition 1.3. Given a closed, oriented n-dimensional smooth manifold N , a one-parameter family of immersions F : N × [0, T ) → Rn+1 is a solution to Inverse Mean Curvature Flow (IMCF) if
∂Ft
∂t (p, t) = 1
Hν(p, t), (p, t) ∈ M × [0, T ), (2)
where H > 0 and ν is the outward unit normal of Nt= Ft(N ).
So the velocity vector at any point on the evolving surface Mtequals −Hν everywhere under MCF, and H1ν everywhere under IMCF.
We are concerned with the initial value problem for these extrinsic flows: given an initial hypersurface M0, how does the solution to either (1) or (2) that has F0(M ) = M0 behave?
One can verify by writing the above equations in components that each one defines a system of second-order nonlinear parabolic or “heat-like” partial differential equations. Therefore, we expect solutions of either flow to exhibit behaviors similar to those of solutions to the heat equation: for instance, instant smoothing and convergence of solutions to something very homogeneous. But due to the nonlinearities in each of these PDE, we also expect certain geometric quantities associated with the evolving surface to become infinitely large in finite- time, thereby terminating the flow before this convergence can take place.
We begin this investigation by determining how a round sphere SR(x0) will behave when evolved by both MCF and IMCF. In this context, both (1) and (2) reduce to an ODE thanks to the spherical symmetry of the solution.
Example 1.4 (Shrinking Spheres under MCF). Let M0 = SR(x0), that is, a round sphere with radius R. The corresponding solution Mt to (1) is also a round sphere Sr(t)(x0) with the time-dependent radius
r(t) =p
R2− 2nt. (3)
Example 1.5 (Expanding Spheres under IMCF). Let N0 = SR(x0). The corresponding solution Nt to (2) is also a round sphere Sr(t)(x0) with time-dependent radius
r(t) = Rent. (4)
So according to (3), an initial sphere of radius R will shrink in size when evolved under MCF until the time T = R2n2 when it has contracted to a point. On the other hand, by (4), a sphere evolved by IMCF will continue to expand toward infinity through all time.
While the analytic and geometric structure of MCF has been the subject of extensive study, far less is known about the structure of IMCF. Therefore, a theme of my dissertation has been to determine properties of IMCF which serve as analogues to well-known results for MCF. Many of these results contrast with those for MCF in fascinating ways. This research will be the focus of this statement.
2 The Formation of Singularities
There is a striking difference between the solution (3) of MCF and the solution (4) of IMCF.
According to (3), a sphere must contract to a point in finite time under MCF, leaving one with no way of continuing the flow thereafter. On the other hand, according to (1.5), a sphere expanding under IMCF is allowed to continue expanding forever. The former is an example of a singularity for a geometric flow.
Definition 2.1. A solution F : M × [0, T ) → Rn+1 of Mean Curvature Flow has a singularity at a time T < +∞ if there does not exist a time ˜T > T and a solution ˜F : M × [0, ˜T ) → Rn+1 to Mean Curvature Flow so that ˜Ft= Ft for each t ∈ [0, T ).
So a solution Mt has a singularity at the time T if one can not extend the flow past this time. This definition for a singularity also applies to IMCF, and more broadly for any extrinsic flow. If a solution has a singularity at some finite time T , we will say T = Tmax, and otherwise Tmax = +∞. We will call {Mt}0≤t≤Tmax the maximal solution to MCF (Resp. IMCF) with initial condition M0.
How can we determine when singularities form under each of these respective flows? For MCF, elementary flow properties give a remarkably general answer to this question. Not only does any solution to (1) neccessarily develop a singularity in finite time, but we can predict the time by which this singularity will occur!
Theorem 2.2 ([Man11], Singularity Formation for MCF). Let M0 ⊂ Rn+1 be any smooth closed hypersurface, and {Mt}0≤t<Tmax the corresponding maximal solution to MCF. Then Tmax< +∞. In fact Tmax< diam(M2n 0)2, where diam(M0) is the diameter of M0 ⊂ Rn+1.
Therefore, we know how long the singularity takes to form based on the diameter of the initial data M0. Is there an analogous statement to the above one for IMCF? We already know that the answer to this question in general is “no”, since Tmax = +∞ for a round sphere.
However, singularities are still known to develop under IMCF for certain other choices of N0, as I will explain in Section 3. In light of this fact and of Theorem 2.2, I proved a similar statement in [Har20b] which applies to any initial N0 that is not homeomorphic to a sphere.
Theorem 2.3 ([Har20b], Singularity Formation and Self-Intersection for IMCF). Let N0 ⊂ Rn+1 be a closed H > 0 hypersurface without spherical topology, and {Nt}0≤t<Tmax the corre- sponding maximal solution to IMCF. Let t∗ = 2n ln(λmaxdiam(N0)), where λmax is the highest positive principal curvature over N0. Then either Tmax < t∗, or a flow surface Nt intersects itself by the time t∗.
So for N0 without spherical topology, we can ensure that either the flow becomes singular or that different regions on the flow surface collide with one another within a prescribed time interval. For many types of initial data, there will be some other way of ruling out the latter possibility, so this result like Theorem 2.2 also offers a way of identifying when singularities form under IMCF as well as how long singularity formation takes to happen.
The proof of Theorem 2.3 utilizes a notion of generalized, or “weak”, solutions of Inverse Mean Curvature Flow first explored by the authors in [HI01]. In this rather complex reformu- lation of IMCF, a weak flow surface ˜Ntof IMCF will coincide with the classical flow surface Nt
until just before Nt becomes singular, at which time ˜Nt will “jump” beyond the surface Nt in space and continue flowing. First, I verified that the only reasons a jump could occur for ˜Nt
would either be an imminent singularity or self-intersection in the classical solution Nt. Then, by proving that these ˜Ntobey a reflection property similar to the one in [CG01], I was able to confine when these jumps can possibly occur to the time 12t∗= ln(λmaxdiam(N0)). This allowed me to also estimate when a singularity or self-intersection must happen by for Nt, if it happens at all. By showing that a jump must always occur for N0 without spherical topology, I obtained the result.
3 The Characterization of Singularities
Now that we understand when singularities form both under MCF and under IMCF, we would like to know what limiting behavior each flow will exhibit near a singular time Tmax. Once again turning to Example 1.4, the principal curvatures λi on a sphere of radius r each equal 1r everywhere, so since the radius r(t) shrinks to 0 as t → Tmax, these curvatures become infinitely large near this time. Like before, this is indicative of more general behavior of MCF. Defining the total curvature |A| of Mtby the function |A|2(p) = Σni=1λ2i(p) over Mt, a singularity of MCF is always characterized by blowup in |A| somewhere on Mt.
Theorem 3.1 ([Man11], Characterizing Singularities of MCF). Let F : M × [0, T ) → Rn+1 be a solution of MCF. Then T = Tmax if and only if limt→Tmaxp∈Mt|A|(p) = +∞.
How does this picture change with IMCF? For this flow, I showed in [Har20c] that the above statement does not hold true for at least one family of initial data.
Theorem 3.2 ([Har20c], Boundedness of Total Curvature under IMCF). Let N0 ⊂ R3 be an H > 0 embedded torus which is rotationally symmetric about its central axis. Then for the corresponding maximal solution {Nt}0≤t<Tmax to (2), Tmax < +∞ and there is a constant C < ∞ so that maxp∈Nt|A|(p) ≤ C for each t ∈ [0, T ).
For this family of solutions, the ring of the expanding torus Ntwhich is closest to the axis of rotation moves toward this point. Using Theorem 2.3, I proved that a singularity does indeed happen, but I also showed that the hole is not completely filled by the singular time Tmax. So the inner ring does not reach the center before Nt goes extinct, and this allows one to derive a uniform bound on |A| up to the singular time.
This is a very surprising result from the point of view of PDE analysis, since the Second Fundamental Form A of Nt is in some sense the second derivative of the surface Nt. Due to the smoothing properties of heat equations, a control on the size of the second derivative near some time T is usually sufficient to ensure the existence a smooth limit function, or in this case a smooth limit surface, for the solution to converge to as t → T . If one continues the evolution from this limit, the solution continues beyond the time T . In this case, the rotationally symmetric torus does approach an embedded limit surface NTmax at the time Tmax, but H will equal 0 at some points along NTmax. Therefore, restarting IMCF from NTmax is impossible since the RHS of (2) would not be defined everywhere.
This is the only example of which I am currently aware where a geometric flow converges to a limit surface at a singular time (without first rescaling the flow surfaces to obtain the limit, that is). Furthermore, it leads one to ask whether |A| remains bounded near any singularity of IMCF. I also provide some first steps toward answering this in [Har20c].
4 The Dynamical Stability of Round Spheres
According to Example 1.4 and Example 1.5, round spheres remain round under both MCF and IMCF, changing only their scale but not their shape as time progresses. This compels us to ask which sort of deformations one could make to a round sphere such that evolving by either (1) or (2) restores the deformed surface to a round sphere. This is an example of a dynamical stability problem in PDE, where in this case we want to understand how stable spheres are under both MCF and IMCF. The first major results in this direction come from [Hui84] and [Ger90].
Theorem 4.1 ([Hui84]). Let M0 ⊂ Rn+1 be a convex hypersurface, and {Mt}0≤t<Tmax be the corresponding maximal solution to MCF. Then the Mt converge in C∞ topology to some point x0 ∈ Rn+1 as t → Tmax. Furthermore, the surfaces ˜Mt = √ 1
Tmax−tMt each rescaled by a factor of √ 1
Tmax−t about x0 converge in C∞ topology to some round sphere SR(x0).
Theorem 4.2 ([Ger90]). Let N0⊂ Rn+1 be a star-shaped hypersurface (W.L.O.G. with respect to the origin), and {Nt}0≤t<Tmax the corresponding maximal solution to IMCF. Then Tmax = +∞, the surfaces ˜Nt= e−ntNteach rescaled by a factor of e−nt about the origin converge in C∞ topology to some round sphere SR(0) as t → +∞.
So, although MCF shrinks a convex surface M0to a point x0 at the singular time, if one were to magnify or “blow up” the flow surfaces Mt through time about x0 by the appropriate scale factors, one would see that MCF is actually making Mt more and more round as it contracts.
Similarly, for star-shaped N0, shrinking or “blowing down” Nt by an appropriate scale factor through time reveals convergence to a sphere.
The interesting difference here is that, where star-shaped is a weaker assumption on a surface than convex, a star-shaped M0 may not converge to a sphere under MCF. So, despite the fact that equation (2) is more nonlinear than equation (1), the sphere is actually more stable under IMCF than it is under MCF. This made me wonder how much more stable, and in what ways.
Rotationally symmetric surfaces, that is, deformations of the sphere which maintain rota- tional symmetry with respect to some axis, provide another rich source of non-round singularities for MCF, see [Hui90], [AAG95], and [AV97]. Whether or not the same is true for IMCF was previously unknown, and so I proved the following Theorem in [Har20a].
Theorem 4.3 ([Har20a], Rotationally Symmetric Stability of Round Spheres). Let N0 ⊂ Rn+1 be an H > 0, embedded, rotationally symmetric surface with spherical topology. Assume further that N0 satisfies an additional curvature condition. Then for the corresponding solution {Nt}0≤t<Tmax to (2), Tmax= +∞ and the rescaled surfaces ˜Nt= e−ntNtconverge in C∞topology to some round sphere SR(x0) as t → ∞.
We do not explain the extra curvature condition here since it is quite technical. There are, however, non-star-shaped examples of rotationally symmetric N0 which satisfy this condition, and so there are also large families of non-star-shaped surfaces which IMCF flows into round spheres according to this result.
I should also remark that there is strong evidence, albeit not yet a conclusive proof, that the curvature condition in this theorem may be dropped. Examining the behavior of the weak solutions detailed in Section 2 over any H > 0, rotationally symmetric embedded sphere N0
strongly suggests that the classical solution Nt should exist for all time and asymptotically converge to a sphere. So Theorem 4.3 is likely true for any rotationally symmetric embedded sphere, which would further confirm that the rounding properties of IMCF are much stronger.
5 Further Directions: Applications in Relativity
With a far clearer picture of the singular and convergence properties of IMCF in hand, my next project will be to study the flow in ambient Riemannian spaces other than Euclidean space.
Specifically, I plan to look at the behavior of IMCF in a family of non-compact Riemannian manifolds known as asymptotically flat manifolds. My interest in these spaces in particular is due to the fact that they model isolated gravitating systems in general relativity.
As an expanding flow, IMCF allows us to relate the intrinsic geometry of an asymptotically flat manifold out at infinity to the geometry of surfaces inside it. This has major applications for problems related to mass in GR. Most all notions of total mass for a spacetime depend on its geometry at infinity, and for this reason IMCF can be used to relate the total mass of a spacetime to the geometry of surfaces lying inside it, such as the event horizon for a black hole.
In fact, the authors in [HI01] developed their weak solutions of IMCF mentioned in Section 2 specifically to prove the Riemannian Penrose Inequality, an inequality which relates the area of a black hole event horizon to total mass, and IMCF has subsequently been used to prove a number of similar mass inequalities. I plan to further explore the role of IMCF in mathematical relativity, and this may open up a broader study of mathematical problems in GR altogether.
References
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