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During this week, I have been studying from the textbook "Geometry of Curves and Surfaces" by do Carmo to learn about the basics of differential geometry and work toward understanding the concepts involved in my research question. The main topics I learned this week were orientability of surfaces, the Gauss map, Gaussian curvature, and the covariant derivative.
During this week, began reading the paper "Singular Limit Laminations, Morse Index, and Positive Scalar Curvature" by Colding and De Lellis because a technique used in that paper is applicable to my project. This technique involves using a skewed metric product to find minimal spheres of a given 3-manifold. As an exercise, I calculated and verified a metric for the 3-ellipse relevant to my research question.
To find the minimal spheres of the 3-ellipse, I will be looking for the geodesic curves of a two-dimensional space created by taking advantage of the rotational symmetry of the ellipse. I calculated the Christoffel symbols for this space based from its metric, which allowed me to create a system of differential equations which describe all geodesic curves of the space. I read multiple papers on related problems to learn about methods used to find geodesic curves in problems similar to this; as I continue working I will try these methods to see how effective they are in this case. I also read some sections of the textbook "Riemannian Manifolds: An Introduction to Curvature" by Lee to learn about the properties of geodesics and some important theorems about them.
This week I created a matlab program to simulate the geodesic curves according to the differenial equations I derived earlier. While the actual version of the metric we are interested in involves a scaling function that is somewhat difficult to work with, in the version I ran on matlab I replaced in with a simple quadratic function which approximates the actual function. This quadratic function, like the original scaling function, contains a parameter which corresponds to the length of the 3-ellipsoid. I tested several different values of this parameter between .01 and .05. The goal of this matlab program was to find geodesic curves that are symmetrical about a central line and hit a certain boundary of the space. In order to find these, I picked an arbitrary point on the central line and had the program compute a geodesic starting at that point with horizontal trajectory, and then shifted the initial point until I found a value where the geodesic hit a boundary. This is a common technique in numerical analysis called the shooting method. What I found was that when the 3-ellipsoid is longer, there are more of these geodesics, which was what we expected to find. I also found that as the length increases, the geodesics which were already present move away from the region in which new geodesics are appearing. Running this program showed some comcrete examples of the phenomenon my research question is trying to prove, and helped me visualize what is happening geometrically.
During this week I came up with a strategy to prove the conjecture by proving several lemmas first. The strategy involves splitting the geodesic curves into separate segments, and showing that the behavior at the end of each segment depends in a continuous way on the behavior at the beginning. The first step will be to show that for any point which lies on the line halfway between the top and bottom of the metric space (the line phi = pi/2), there is some angle such that the geodesic starting at that point with a velocity vector at that angle will hit the edge of the metric space before it passes through phi = pi/2 again. The argument for proving this will have to use the fact that some large initial angles will cause the geodesic to miss the edge due to curving toward the right, and some small initial angles will cause the geodesic to miss the edge due to curving to the left. If I can show that the behavior of the geodesic varies continuously with respect to the initial angle, then I can show that there is an angle that makes the geodesic hit the edge. I am currently working on a proof of this step; so far I have shown that curves of the first type can always be found, but I haven't been able to show yet that curves of the second type can always be found. The second step will be to show that if a geodesic curve begins on the line phi = pi/2 and returns to it, the angle of the velocity vector at the end of the curve depends continuously and monotonically on the angle of the velocity vector at the beginning of the curve. The third step will be to show that if a geodesic curve begins at a point (r_0, 0) with a velocity vector pointing straight in the phi direction, the angle of the velocity vector when the geodesic first crosses the line phi = pi/2 depends continuously and monotonically on r_0. These three lemmas, along with some properties about how the behavior changes as the scaling function lambda changes, should give the result I'm trying to prove. All of these lemmas would confirm properties I have seen in the matlab simulation. I got the idea of parametrizing geodesics by angle of the velocity vector, and observing change in behavior when modifying initial conditions, from a paper entitled "Shrinking Donuts" by Angenent.
During this week I began to work on proofs of the lemmas I need to prove before I can prove the main theorem, and to write up a typed version of these proofs. I have almost completed the proof of the first lemma, and I have begun to work on some of the others. Methods from the Angenent paper continue to be useful in this problem. In some of these lemmas it has been useful to apply a bound on what region of the metric space I am looking at, because in some regions the behavior of the geodesics is easier to predict than in others. For example, if the behavior of a geodesic depends on whether a function is positive or negative, and this function depends on multiple variables, it is much easier to predict the sign of the function in a region where the change provided by each variable moves the function in the same direction than in a region where the changes provided by the different variables more the function in opposite directions. In the latter case it becomes necessary to calculate partial derivatives, and with these nonlinear equations it sometimes becomes difficult to predict behavior. Making these restrictions on what region of the metric space I'm looking at will not make the theorem less general, because as lambda approaches 1 this region takes up more and more of the metric space. Currently, I've found there are two major problems with the lemmas I've constructed so far which will require me to change them a bit. The first is that these lemmas only describe the change in the angle of the unit tangent vector between two points, and don't describe how shifting the initial angle causes the final point to move, which will need to be a major part of the argument. The second is that I've been using the derivative of this velocity angle with respect to arc length as a way to compare the curvature of different geodesics, without taking into account that the same change in r or phi can result in a very different arc length in different parts of the metric space due to the nonlinear properties of the metric. Overall, I think I've made some major progress this week with regard to the overall structure of the proof, and I hope to fix some of its problems soon.
During this week I began to resolve some of the problems I noticed in my arguments last week, and to construct arguments for other parts of the proof. While I was talking to my mentor about the struggles I was having with the proof, he suggested that I find ways to use the fact that epsilon is arbitrarily small in my arguments, rather than letting it be arbitrary and then taking the limit as it goes to zero. This ended up being helpful in one of the lemmas I had been unable to complete. Another challenge I had in completing that lemma was that it required comparing the effects of two opposite effects on the solution curves to a set of equations, and until this week I hadn't been able to find a manageable way to show under what conditions one of the two effects dominates. Another area of the proof in which I made progress was a section that concerns the behavior of geodesic curves which begin and end at the line phi=pi/2. Understanding the behavior of geodesics like these is a major piece of the proof; what I need to do is show how the position and angle of the curve when it returns to phi=pi/2 relates to its initial angle when it left the line. The insight I had was to divide each geodesic into three segments based on concavity; each of these segments is easy to study on its own, and seems like a shift in one segment causes an easily predictable shift in the next segments. This part of the proof is not complete yet but I believe I can finish it soon. Another part of the proof in which I made progress was a lemma I have been trying to prove for weeks, stating that each point along the line phi=pi/2 has exactly one "critical angle," which is an angle at which a geodesic would need to proceed from that starting point such that it would immediately run into a certain boundary of the metric space. I came up with a strategy to show this, and have a nearly complete proof of it. This is really helpful, because several other facts I was trying to show follow from this lemma. Overall, I have been making progress in several of the different component arguments that make up the proof, but many of these are still incomplete.
During this week, I put a lot of time into writing my paper and presenting my results in an understandable way. While doing so, I encountered some flaws in some of my reasoning, which in some cases I was able to fix but in others required removing sections from my paper until I am able to fix them. I have a clearer idea now about the next steps which will be necessary to continue my work on this theorem. For example, I will need to show that at a given arc length along a geodesic, the value of the rate of change of the velocity angle at that point changes continuously and non-constantly with respect to the initial angle of the geodesic. In addition to this, I will need to get a deeper understanding of how this rate of change changes with respect to the position of a geodesic in the metric space. Overall, I didn't make as much progress toward proving this theorem as I had initially hoped, but I'm content with what I was able to get done in these two months and am confident that I will be able to build what I have now into a complete proof.
During these final days of the REU I continued editing my paper and working on my presentation. Following the advice of my mentor, I tried to make the presentation as understandable as possible to people unfamiliar with differential geometry, put more diagrams and images in the presentation than equations and technicalities. Now that the program is ending, I plan on continuing to work on this project with the hope of finishing the proof of the theorem.