arXiv subjects

Jorge Lira

Publications and source records attributed to Jorge Lira.

5 recordsLinked to original sources

arXiv2022

Mean curvature flow of graphs in Generalized Robertson-Walker spacetimes with perpendicular Neumann boundary condition

We prove the longtime existence for the mean curvature flow problem with a perpendicular Neumann boundary condition in a Generalized Robertson Walker (GRW) spacetime that obeys the null convergence condition. In addition, we prove that the metric of such a solution is conformal to the one of the leaf of the GRW in asymptotic time. Furthermore, if the initial hypersurface is mean convex, then the evolving hypersurfaces remain mean convex during the flow.

math.DG↗

arXiv2022

Einstein Type Systems on Complete Manifolds

In this paper, we study the coupled Einstein constraint equations on complete manifolds through the conformal method, focusing on non-compact manifolds with flexible asymptotics. This is physically well-motivated by standard cosmological space-times with non-compact Cauchy hypersurfaces, which favour general bounded geometry manifolds rather than a specific model for infinity. First, we prove an existence criterion on complete manifolds with appropriate barrier functions for physically well-motivated coupled systems. Then, in the bounded geometry case, we build barrier functions and thus show existence. We also prove an existence result on compact manifolds with boundary for a wider family of coupled systems.

math.AP↗

arXiv2021

On the Positive Energy Theorem for Stationary Solutions to Fourth-Order Gravity

In this paper we prove a positive energy theorem related to fourth-order gravitational theories, which is a higher-order analogue of the classical ADM positive energy theorem of general relativity. We will also show that, in parallel to the corresponding situation in general relativity, this result intersects several important problems in geometric analysis. For instance, it underlies positive mass theorems associated to the Paneitz operator, playing a similar role in the positive $Q$-curvature conformal prescription problem as the Schoen-Yau positive energy theorem does for the Yamabe problem. Several other links to $Q$-curvature analysis and rigidity phenomena are established.

math.DG↗

arXiv2021

Short-time existence for the network flow

This paper contains a new proof of the short-time existence for the flow by curvature of a network of curves in the plane. Appearing initially in metallurgy and as a model for the evolution of grain boundaries, this flow was later treated by Brakke \cite{Br} using varifold methods. There is good reason to treat this problem by a direct PDE approach, but doing so requires one to deal with the singular nature of the PDE at the vertices of the network. This was handled in cases of increasing generality by Bronsard-Reitich \cite{BrRe}, Mantegazza-Novaga-Tortorelli \cite{MNT} and eventually, in the most general case of irregular networks by Ilmanen-Neves-Schulze \cite{INS}. Although the present paper proves a result similar to the one in \cite{INS}, the method here provides substantially more detailed information about how an irregular network `resolves' into a regular one. Either approach relies on the existence of self-similar expanding solutions found in \cite{MS}. As a precursor to and illustration of the main theorem, we also prove an unexpected regularity result for the mixed Cauchy-Dirichlet boundary problem for the linear heat equation on a manifold with boundary.

math.DG↗

arXiv2020

Visualizing and Understanding Large-Scale Assessments in Mathematics through Dimensionality Reduction

In this paper, we apply the Logistic PCA (LPCA) as a dimensionality reduction tool for visualizing patterns and characterizing the relevance of mathematics abilities from a given population measured by a large-scale assessment. We establish an equivalence of parameters between LPCA, Inner Product Representation (IPR) and the two paramenter logistic model (2PL) from the Item Response Theory (IRT). This equivalence provides three complemetary ways of looking at data that assists professionals in education to perform in-context interpretations. Particularly, we analyse the data collected from SPAECE, a large-scale assessment in Mathematics that has been applied yearly in the public educational system of the state of Ceará, Brazil. As the main result, we show that the the poor performance of examinees in the end of middle school is primarily caused by their disabilities in number sense.

stat.AP↗