arXiv subjects
Publications and source records attributed to Hongda Qiu.
We construct compact convex bodies whose boundaries contain a nontrivial geodesic segment and are strictly convex elsewhere. The construction applies if sectional curvature is constant near a point, or if a normal Jacobi operator has a simple maximal eigenvalue. In particular, such a body exists in every smooth three-dimensional or four-dimensional Riemannian manifold.
We construct a closed surface of genus $2$ with nonpositive Gaussian curvature and a complete nonperiodic flat geodesic asymptotic to a flat closed geodesic. The metric is of class $C^{3,1/3}$ and its curvature is of class $C^{1,1/3}$, but not $C^2$. Thus the periodic-asymptote obstruction for smooth surfaces does not extend to $C^1$ Gaussian curvature.
We construct a smooth embedded annulus in Euclidean 3-space with negative Gauss curvature in its interior whose boundary consists of strictly convex curves in planes tangent to the annulus, and which contains closed asymptotic curves. This gives a counterexample to a 1962 conjecture of Nirenberg. The construction is based on Lelieuvre's formula, which yields a closed curve that is then realized as an asymptotic curve on a ruled annulus. The second asymptotic curve is obtained from a periodic solution of a Riccati equation.
We show that the maximal number of nonsimultaneous elastic collisions among $n$ identical hard disks moving freely on the plane is at least exponential in $n$.
We prove that if a topological sphere smoothly embedded into $\mathbb{R}^3$ with normal curvatures absolutely bounded by $1$ is contained in an open ball of radius $2$, then the region it bounds must contain a unit ball. This result suggests a potential direction for a problem formulated by D.Burago and A.Petrunin asking whether a topological sphere smoothly embedded in $\mathbb{R}^3$ with normal curvatures absolutely bounded by $1$ encloses a volume of at least $\frac{4}{3}π$. The appendix presents an example illustrating an alternative aspect for this problem.
Let $(M,g)$ be a $C^\infty$-smooth, $n$-dimensional Riemannian manifold which is diffeomorphic to $\RR^n$ and admit an action of a properly discontinuous and cocompact group. This work proves the existence of a $C^\infty$ equivariant isometric embedding of $M$ in some Euclidean space $\RR^q$ where $q = \max\{s_n+2n, s_n+n+5\}$ is the same as the dimension of Matthias Günther's results.
This note pertains to isometric embeddings endowed with certain geometric properties. We study two embedding problems for a Riemannian manifold $M$ which is diffeomorphic to $\RR^n$ and admits a Bieberbach group $Γ$ acting by isometries. The first problem concerns the existence of an isometric embedding of $M$ into a bounded subset of some Euclidean space $\RR^{D_1}$. The second problem seeks a $Γ$-equivariant isometric embdding of $M$ into $\RR^{D_2}$. By using a known trick in a novel way, our idea yields results with $D_1 = N+2n$ and $D_2 = N+n$, where $N$ is the Nash dimension of $ M/Γ$. Moreover, we also show that an $n$-dimensional smooth manifold, of Nash dimension $N$, can be isometrically embedded into a bounded subset of $\RR^{2N}$.
This work pose an example of a smooth closed surface in $\mathbb{R}^3$ which has genus $0$, normal curvatures at most $1$ in absolute value and encloses a volume smaller than the volume of a unit ball. It gives a negative answer to a question asked by Dmitri Burago and Anton Petrunin.
Interacting agent and particle systems are extensively used to model complex phenomena in science and engineering. We consider the problem of learning interaction kernels in these dynamical systems constrained to evolve on Riemannian manifolds from given trajectory data. The models we consider are based on interaction kernels depending on pairwise Riemannian distances between agents, with agents interacting locally along the direction of the shortest geodesic connecting them. We show that our estimators converge at a rate that is independent of the dimension of the state space, and derive bounds on the trajectory estimation error, on the manifold, between the observed and estimated dynamics. We demonstrate the performance of our estimator on two classical first order interacting systems: Opinion Dynamics and a Predator-Swarm system, with each system constrained on two prototypical manifolds, the $2$-dimensional sphere and the Poincaré disk model of hyperbolic space.
We develop a new framework for multi-agent collision avoidance problem. The framework combined traditional pathfinding algorithm and reinforcement learning. In our approach, the agents learn whether to be navigated or to take simple actions to avoid their partners via a deep neural network trained by reinforcement learning at each time step. This framework makes it possible for agents to arrive terminal points in abstract new scenarios. In our experiments, we use Unity3D and Tensorflow to build the model and environment for our scenarios. We analyze the results and modify the parameters to approach a well-behaved strategy for our agents. Our strategy could be attached in different environments under different cases, especially when the scale is large.