Deformation of Scalar Curvature and Volume

The stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the Einstein metrics. One may consider the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equal to a given constant. In this paper, we localize a condition satisfied by such stationary points to smooth bounded domains. The condition involves a generalization of the static equations, and we interpret solutions (and their boundary values) of this equation variationally. On domains carrying a metric that does not satisfy the condition, we establish a local deformation theorem that allows one to achieve simultaneously small prescribed changes of the scalar curvature and of the volume by a compactly supported variation of the metric. We apply this result to obtain a localized gluing theorem for constant scalar curvature metrics in which the total volume is preserved. Finally, we note that starting from a counterexample of Min-Oo's conjecture such as that of Brendle-Marques-Neves, counterexamples of arbitrarily large volume and different topological types can be constructed.Comment: All comments welcome! Published version: Math. Ann. (to appear

Extension of a theorem of Shi and Tam

In this note, we prove the following generalization of a theorem of Shi and Tam \cite{ShiTam02}: Let $(\Omega, g)$ be an $n$-dimensional ($n \geq 3$) compact Riemannian manifold, spin when $n>7$, with non-negative scalar curvature and mean convex boundary. If every boundary component $\Sigma_i$ has positive scalar curvature and embeds isometrically as a mean convex star-shaped hypersurface ${\hat \Sigma}_i \subset \R^n$, then \int_{\Sigma_i} H d \sigma \le \int_{{\hat \Sigma}_i} \hat{H} d {\hat \sigma} where $H$ is the mean curvature of $\Sigma_i$ in $(\Omega, g)$, $\hat{H}$ is the Euclidean mean curvature of ${\hat \Sigma}_i$ in $\R^n$, and where $d \sigma$ and $d {\hat \sigma}$ denote the respective volume forms. Moreover, equality in (\ref{eqn: main theorem}) holds for some boundary component $\Sigma_i$ if, and only if, $(\Omega, g)$ is isometric to a domain in $\R^n$. In the proof, we make use of a foliation of the exterior of the $\hat \Sigma_i$'s in $\R^n$ by the $\frac{H}{R}$-flow studied by Gerhardt \cite{Gerhardt90} and Urbas \cite{Urbas90}. We also carefully establish the rigidity statement in low dimensions without the spin assumption that was used in \cite{ShiTam02}Comment: Shortened title and revised. To appear in Calculus of Variations and PDE'

Dynamics of shape fluctuations of quasi-spherical vesicles revisited

In this paper, the dynamics of spontaneous shape fluctuations of a single, giant quasi-spherical vesicle formed of a single lipid species is revisited theoretically. A coherent physical theory for the dynamics is developed based on a number of fundamental principles and considerations and a systematic formulation of the theory is also established. From the systematic theoretical formulation, an analytical description of the dynamics of shape fluctuations of quasi-spherical vesicles is derived. In particular, in developing the theory we have made a new interpretation of some of the phenomenological constants in a canonical continuum description of fluid lipid-bilayer membranes and shown the consequences of this new interpretation in terms of the characteristics of the dynamics of vesicle shape fluctuations. Moreover, we have used the systematic formulation of our theory as a framework against which we have discussed the previously existing theories and their discrepancies. Finally, we have made a systematic prediction about the system-dependent characteristics of the relaxation dynamics of shape fluctuations of quasi-spherical vesicles with a view of experimental studies of the phenomenon and also discussed, based on our theory, a recently published experimental work on the topic.Comment: 18 pages, 4 figure

Socially Aware Motion Planning with Deep Reinforcement Learning

For robotic vehicles to navigate safely and efficiently in pedestrian-rich environments, it is important to model subtle human behaviors and navigation rules (e.g., passing on the right). However, while instinctive to humans, socially compliant navigation is still difficult to quantify due to the stochasticity in people's behaviors. Existing works are mostly focused on using feature-matching techniques to describe and imitate human paths, but often do not generalize well since the feature values can vary from person to person, and even run to run. This work notes that while it is challenging to directly specify the details of what to do (precise mechanisms of human navigation), it is straightforward to specify what not to do (violations of social norms). Specifically, using deep reinforcement learning, this work develops a time-efficient navigation policy that respects common social norms. The proposed method is shown to enable fully autonomous navigation of a robotic vehicle moving at human walking speed in an environment with many pedestrians.Comment: 8 page

Configurable Distributed Physical Downlink Control Channel for 5G New Radio: ResourceBundling and Diversity Trade-off

New radio technologies for the fifth generation of wireless system have been extensively studied globally. Specifically, air interface protocols for 5G radio access network will be standardized in coming years by 3GPP. Due to its crucial function in scheduled system, physical layer downlink control channel (PDCCH) is a core element to enable all physical layer data transmissions. Recently, configurable distributed PDCCH with the intention to cope with different scenarios has been developed in 3GPP. To have comprehensive understanding of respective technical advantages and potential scenario dependent limitations, detailed performance analysis and evaluations of configurable distributed PDCCH are thoroughly studied in this paper. In particular, exponential effective SNR mapping (EESM) has been employed as the performance metric of configurable distributed PDCCH in different scenarios. It is demonstrated from EESM results that configurable distributed PDCCH offers additional degree of freedom for the trade-off between achieved frequency diversity and channel estimation gain by adjusting resource bundling level according to the channel and interference scenario experienced by the control channel transmission

On the Bartnik extension problem for the static vacuum Einstein equations

We develop a framework for understanding the existence of asymptotically flat solutions to the static vacuum Einstein equations with prescribed boundary data consisting of the induced metric and mean curvature on a 2-sphere. A partial existence result is obtained, giving a partial resolution of a conjecture of Bartnik on such static vacuum extensions. The existence and uniqueness of such extensions is closely related to Bartnik's definition of quasi-local mass.Comment: 33 pages, 1 figure. Minor revision of v2. Final version, to appear in Class. Quantum Gravit