Abstract
This dissertation presents an investigation of privileged geometric surfaces in spacetimes, with a focus on two classes of extremal surfaces: marginally trapped surfaces which foliate an apparent horizon and bulge surfaces in the Python's Lunch Conjecture. These geometric objects serve as essential tools for understanding gravitational physics in the modern era of holography.
The first major contribution concerns the apparent horizons. Unlike event horizons, which require global knowledge of spacetime evolution, apparent horizons can be identified by only using the local geometry. We establish the existence of locally spacelike apparent horizons near any Hubeny-Rangamani-Takayanagi (HRT) surface in spherically symmetric, asymptotically AdS spacetimes satisfying the Null Energy Condition. Our analysis reveals that the area of marginal surfaces comprising these horizons increases monotonically away from the HRT surface, connecting holographic entanglement entropy with traditional gravitational trapping regions. We further demonstrate that spacelike apparent horizons are foliated by minimar surfaces--a specialized class of marginal surfaces that are also minimal on partial Cauchy slices.
The second major contribution involves a detailed investigation of bulge surfaces--extremal surfaces with Morse index 1. These surfaces are conjectured to control the exponential complexity of reconstructing bulk operators in black hole interiors. We establish that the physics-motivated definition of bulges fits naturally within Almgren-Pitts min-max theory and demonstrate several surprising properties that distinguish them from their index-0 counterparts. Most notably, we prove that bulges can spontaneously break spatial isometries of the bulk geometry. For black brane interiors with planar symmetry, this leads to reconstruction complexity that is not extensive in boundary volume, suggesting these interiors are "simpler" to reconstruct than naively expected.
Our analysis also reveals that bulge surfaces are sensitive to compact extra dimensions--an effect absent for minimal surfaces. In AdS_3xS^1, the singular cross-shaped bulge of pure AdS_3 resolves into a smooth second Scherk surface that genuinely probes the extra dimension. For multi-boundary wormhole states, we discover a plateauing phenomenon where adding boundaries beyond a certain threshold provides no computational advantage for interior reconstruction. Through systematic exploration of various spacetime geometries--including vacuum AdS spacetimes, excited states, orbifolds, LLM geometries, and multi-boundary configurations--we provide extensive data for testing the Python's Lunch Conjecture.
Together, these contributions advance our understanding of the geometric properties of spacetimes and clarify their role in the entropic and computational structure of black hole interiors in holography.