arXiv subjects

Alejandro Peñuela Diaz

Publications and source records attributed to Alejandro Peñuela Diaz.

6 recordsLinked to original sources

arXiv2026

Area-constrained Willmore surfaces in Lorentzian manifolds: curvature inequalities and the Hawking quasi-local energy

We develop an area-constrained Willmore theory for closed spacelike surfaces in four-dimensional Lorentzian manifolds. Since the Euler--Lagrange equation is normal-bundle-valued, we introduce directional criticality along a prescribed normal direction and identify a class of connection-compatible directions determined by the normal connection. Our main result is the sharp inequality $2λ|Σ|+\int_Σ|\vec H|^2,dμ\leq 8πχ(Σ)$ for criticality in a connection-compatible spacelike direction under an Einstein-tensor sign condition. Under the dominant energy condition and $λ\geq0$, this yields $\int_Σ|\vec H|^2,dμ\leq16π$. Equality is rigid: $Σ$ is round and satisfies Minkowski causal-diamond rigidity. We also prove an analogous sharp inequality for null critical directions. These results give positivity, rigidity, stationarity, and infinitesimal monotonicity statements for the Hawking quasi-local energy. A hypersurface formulation recovers the previously studied Hawking-surface problem as criticality in the spatial normal direction. Explicit examples in spherical symmetry, Kerr, and FLRW spacetimes distinguish directional from full criticality and show the necessity of the sign assumptions in the rigidity theory.

math.DG↗

arXiv2026

Curvature inequalities and rigidity for constant mean curvature and spacetime constant mean curvature surfaces

We establish curvature inequalities and rigidity results for surfaces satisfying constant mean curvature type conditions in both Riemannian and Lorentzian geometry. In the Riemannian setting, we study constant mean curvature (CMC) surfaces. Building on the Christodoulou-Yau inequality $H^2\leq 16π/ |Σ|$ (with $H$ the mean curvature and $|Σ|$ the area) for CMC surfaces on three-dimensional manifolds with nonnegative scalar curvature, we show that the inequality holds under a weaker stability condition controlling only the constant mode of the second variation. Combined with an extrinsic curvature sign condition, equality forces the region enclosed by the surface to be Euclidean. These results extend to higher dimensions and to the hyperbolic and spherical settings. In the Lorentzian setting, we introduce a stability theory for spacetime constant mean curvature (STCMC) surfaces and prove the sharp inequality $|\vec{H}|^2\leq 16π/ |Σ|$ under the dominant energy condition. We also obtain rigidity for the equality case: under suitable geometric assumptions, the maximal globally hyperbolic development of the enclosed spacelike region is isometric to a causal diamond in Minkowski spacetime. In particular, this implies positivity and rigidity for the Hawking quasi-local energy in the general spacetime setting when evaluated on stable STCMC surfaces. Finally, we analyze the known STCMC foliations in the spacelike and null settings. We show that asymptotic leaves are stable under positive mass conditions, whereas the local matter density and shear govern the instability of local foliations.

math.DG↗

arXiv2026

Rigidity and positivity of Hawking quasi-local energy on area-constrained critical surfaces

A key test for any quasi-local energy in general relativity is that it be nonnegative and satisfy a rigidity property; if it vanishes, the region enclosed is flat. We show that the Hawking energy, also known as the Hawking mass, satisfies these properties under the dominant energy condition when evaluated on its natural area-constrained critical surfaces within a spacelike hypersurface (initial data set). In the time-symmetric case, these critical surfaces coincide with area-constrained Willmore surfaces, and we obtain positivity and rigidity theorems for the Hawking energy on such surfaces, including charged and cosmological constant (hyperbolic and spherical) variants as well as higher-dimensional analogues. In the fully dynamical (non-time-symmetric) case, we establish the first nonnegativity and rigidity theorems for the Hawking energy in this general setting. These results confirm the Hawking energy's consistency with basic physical principles and address several longstanding ambiguities and criticisms.

math-ph↗

arXiv2025

Foliations by critical surfaces of the Hawking energy in asymptotically flat initial data sets

Area-constrained critical surfaces for the Hawking quasi-local energy ("Hawking surfaces") provide a natural setting for that energy: they enjoy positivity and rigidity properties. We construct large-scale foliations at infinity by Hawking surfaces in asymptotically Schwarzschild initial data sets. Using a Lyapunov-Schmidt reduction within a Willmore-foliation framework, we prove existence and uniqueness of the foliation and study its coordinate center. Under the dominant energy condition, we show that along the leaves of the foliation, the Hawking energy is positive and converges to the ADM energy in the large-sphere limit; moreover, subject to an explicit integral constraint, it is monotone along the foliation. Under weaker assumptions we construct an on-center family of Hawking surfaces that, while not necessarily a foliation, still enjoys positivity and the large-sphere limit. Finally, we obtain a rigidity statement and verify that our hypotheses hold in a broad class of data, initial data sets with harmonic or York asymptotics, thereby demonstrating the robustness of Hawking surfaces as a quasi-local energy tool in dynamical spacetimes.

math.DG↗

arXiv2025

Local space time constant mean curvature and constant expansion foliations

Inspired by the small sphere-limit for quasi-local energy we study local foliations of surfaces with prescribed mean curvature. Following the strategy used by Ye in 1991 to study local constant mean curvature foliations, we use a Lyapunov Schmidt reduction in an n+1 dimensional manifold equipped with a symmetric 2-tensor to construct the foliations around a point, prove their uniqueness and show their nonexistence conditions. To be specific, we study two foliation conditions. First we consider constant space-time mean curvature surfaces. These foliations were used by Cederbaum and Sakovich to characterize the center of mass in general relativity. Second, we study local foliations of constant expansion surfaces.

math.DG↗

arXiv2023

Constructing electrically charged Riemannian manifolds with minimal boundary, prescribed asymptotics, and controlled mass

In 2015, Mantoulidis and Schoen constructed $3$-dimensional asymptotically Euclidean manifolds with non-negative scalar curvature whose ADM mass can be made arbitrarily close to the optimal value of the Riemannian Penrose Inequality, while the intrinsic geometry of the outermost minimal surface can be ``far away'' from being round. The resulting manifolds, called extensions, are geometrically not ``close'' to a spatial Schwarzschild manifold. This suggests instability of the Riemannian Penrose Inequality. Their construction was later adapted to $n+1$ dimensions by Cabrera Pacheco and Miao. In recent papers by Alaee, Cabrera Pacheco, and Cederbaum and by Cabrera Pacheco, Cederbaum, and McCormick, a similar construction was performed for asymptotically Euclidean electrically charged Riemannian manifolds and for asymptotically hyperbolic Riemannian manifolds, respectively, obtaining $3$-dimensional extensions. This paper combines and generalizes all the aforementioned results by constructing suitable asymptotically hyperbolic or asymptotically Euclidean extensions with electric charge in $n+1$ dimensions for $n\geq2$. We study in detail the sub-extremality of these manifolds, consider the so far unstudied case of extremality in extensions with electric charge and allow more general conditions for the metric of our extensions. Besides suggesting instability of a naturally conjecturally generalized Riemannian Penrose Inequality, the constructed extensions give insights into an ad hoc generalized notion of Bartnik mass, similar to the Bartnik mass estimate for minimal surfaces proven by Mantoulidis and Schoen via their extensions, and unifying the Bartnik mass estimates in the various scenarios mentioned above.

math.DG↗