---
title: Codimension-Two Bulk Extremal Surfaces
url: https://www.emergentmind.com/topics/codimension-two-bulk-extremal-surfaces
type: topic
---

# Codimension-Two Bulk Extremal Surfaces

A codimension-two bulk extremal surface is a (typically spacelike) submanifold of codimension two that extremizes a geometric functional—most commonly area—in a Lorentzian or pseudo-Riemannian ambient space. Such surfaces play a pivotal role in Lorentzian geometry, the mathematics of submanifold theory, and modern theoretical physics, especially in contexts where extrinsic curvature, notions of umbilicity, and geometric classification of surfaces are required.

## 1. Geometric Construction in Lorentz–Minkowski Space

The systematic study of spacelike codimension-two surfaces in Lorentz–Minkowski space $\mathbb{R}^{n+1}_1$ requires tools that go beyond classical Gauss maps, which are naturally tied to codimension-one (hypersurface) settings. In this context, the notion of an $HS_r$-valued Gauss map provides a canonical method for assigning normal data to each point on such a surface [1102.2527].

Given a spacelike surface $M$ with codimension two, the normal plane $N_p M$ at each $p \in M$ is a timelike 2-plane. Fixing a base point $v = (0, \ldots, 0, -1)$, one considers the model hyperbolic space of radius one,
$$
H^n(v,1) = \{x \in \mathbb{R}^{n+1}_1 : (x-v, x-v) = -1,\, x_{n+1} \geq 0\}
$$
where $(\cdot,\cdot)$ denotes the Lorentzian dot product. Translated normal planes (passing through the origin in this frame) intersect this hyperbolic space in a hyperbola.

For each fixed $r > 0$, introduce the hyperplane $\Pi_r = \{x \in \mathbb{R}^{n+1}_1 : x_{n+1} = r\}$. The intersection $HS_r := H^n(v,1) \cap \Pi_r$ consists of two points for each $p$, denoted $\mathbf{n}_r^+(p)$ and $\mathbf{n}_r^-(p)$. The pair of maps
$$
\mathbf{n}_r^+,\, \mathbf{n}_r^- : M \to HS_r
$$
are then called the $n_r^{\pm}$-Gauss maps. These maps provide a replacement for the unit normal in the higher-codimension setting and are shown to be smooth and well-defined via a local implicit function theorem applied to a specific system of algebraic constraints.

## 2. Differential Geometry and Fundamental Curvatures

Once defined, the $n_r^{\pm}$-Gauss maps yield a powerful framework for analyzing the extrinsic geometry of $M$. Their differentials decompose into tangential and normal components:
$$
d\mathbf{n}_r(p) = (d\mathbf{n}_r)^{\top}(p) + (d\mathbf{n}_r)^{N}(p)
$$
The associated Weingarten map is
$$
A^{n_r} = - (d\mathbf{n}_r)^{\top}: T_p M \to T_p M
$$
which is symmetric. At $p$, its eigenvalues, denoted $k_1^{n_r}(p),\dotsc, k_{n-1}^{n_r}(p)$, serve as the $n_r$-principal curvatures. The corresponding curvatures are:
- $n_r$-Gauss–Kronecker curvature: $K_p^{n_r} = \det(A^{n_r})$
- $n_r$-mean curvature: $H_p^{n_r} = \frac{1}{n-1}\operatorname{tr}(A^{n_r})$

A point $p$ is called $n_r$-umbilic if all $n_r$-principal curvatures coincide, and $n_r$-flat if they all vanish. When $M$ is $n_r$-umbilic (respectively, $n_r$-flat) at every point for every $r > 0$, it is termed totally umbilic (resp., totally flat).

## 3. Characterization of Flat and Umbilic Surfaces

The $n_r^{\pm}$-Gauss maps offer analytic criteria linking the constancy and geometric properties of these maps to the global structure of $M$ [1102.2527]:
- $M$ is $n_r$-flat for some $r$ if and only if $\mathbf{n}_r^+$ or $\mathbf{n}_r^-$ is constant. This is equivalent to $M$ being contained in an affine hyperplane.
- If $M$ is $n_r$-umbilic for some $r$ and subject to appropriate geometric constraints (e.g., lying in a pseudo-hypersphere), it is actually totally umbilic. In particular, umbilicity relative to two independent normal fields (e.g., both $n_r^+$ and $n_r^-$) implies umbilicity relative to any smooth normal field, simplifying classification.

Precise theorems formalize these statements. For instance, in the context of $M$ contained in a hyperbolic space $H^n(0,R)$, the equivalence of $n_r$-umbilicity, total umbilicity, and containment in a hyperplane is established.

## 4. Codimension-Two Extremal Surfaces and Broader Significance

In higher-codimension Lorentzian geometry and physical theories (notably, general relativity and holography), codimension-two extremal surfaces are integral to the analysis of bulk geometry. They arise naturally when studying:
- Variational problems such as hypersurfaces (minimal or extremal area) in time-oriented spacetimes,
- The Ryu–Takayanagi and Hubeny–Rangamani–Takayanagi formulas for entanglement entropy, where the surfaces in question are codimension-two and must be extremal in the bulk,
- Problems in rigidity, stability, and classification of globally or locally extreme objects under geometric flows or constraints.

The $\mathbf{n}_r^{\pm}$-Gauss maps furnish invariant curvature quantities defined extrinsically via the ambient Lorentzian geometry, facilitating rigorous criteria for flatness and umbilicity that are valid even when classical codimension-one tools are insufficient.

Moreover, since these Gauss maps target a hyperbolic space, they link the study of such surfaces with other geometric settings where hyperbolic geometry is fundamental, which is common in relativistic and holographic applications.

## 5. Main Formulas and Computational Framework

A concise summary of the foundational mathematical objects and formulas:

- **Model hyperbolic space**:
  $$
  H^n(v, 1) = \{ x \in \mathbb{R}^{n+1}_1 : (x - v, x - v) = -1,\, x_{n+1} \geq 0 \}
  $$
- **Intersection hyperplane**:
  $$
  \Pi_r = \{ x \in \mathbb{R}^{n+1}_1 : x_{n+1} = r \}
  $$
- **$n_r^{\pm}$-Gauss maps**:
  $$
  n_r^+, n_r^- : M \to HS_r = H^n(v, 1) \cap \Pi_r
  $$
- **Weingarten map and principle curvatures**:
  $$
  A^{n_r} = - (dn_r)^{\top}, \quad \text{eigenvalues: } k_i^{n_r}(p)
  $$
- **Curvature invariants**:
  $$
  K_p^{n_r} = \det(A^{n_r}), \quad H_p^{n_r} = \frac{1}{n-1} \operatorname{tr}(A^{n_r})
  $$

In local coordinates, the points $a = (a_1, \dots, a_n, r)$ representing $n_r^{\pm}(p)$ are determined by the linear system:
\[
\begin{aligned}
& (X_{u_i}(p), a - v) = 0 \quad \text{for } i = 1, \dots, n-1, \\
& (a - v, a - v) = -1.
\end{aligned}
\]
Here $X(u)$ is a local parametrization of $M$ and $\{ X_{u_1}, \dots, X_{u_{n-1}} \}$ spans the tangent space.

## 6. Applications and Contextual Significance

The introduction of $n_r^{\pm}$-Gauss maps enables a complete characterization of flat and umbilic codimension-two surfaces in Lorentz–Minkowski space. This has ramifications for:
- Analytic description and classification of physically significant surfaces in mathematical relativity,
- The study of bulk extremal surfaces in AdS/CFT and related dualities, especially where higher codimension or nontrivial extrinsic geometry is involved,
- Generalizations to variational problems for area, Willmore energy, and related functionals in pseudo-Riemannian settings.

By encoding curvature data in the behavior of $n_r^{\pm}$, one obtains a bridge from local geometric invariants to overarching extrinsic structures (such as totally umbilic or totally flat surfaces) and associated global embedding problems.

## 7. Summary Table: Core Geometric Structures

| Structure                | Definition/Construction                                 | Key Role                                      |
|--------------------------|--------------------------------------------------------|------------------------------------------------|
| $H^n(v,1)$               | Model hyperbolic space in $\mathbb{R}^{n+1}_1$         | Target for Gauss map; ambient normal geometry  |
| $HS_r$                   | $H^n(v,1) \cap \{x_{n+1}=r\}$                          | Range of $n_r^{\pm}$-Gauss map                 |
| $n_r^{\pm}$-Gauss maps   | Intersection points of translated normal and $HS_r$    | Encodes extrinsic normals at each $p \in M$    |
| $A^{n_r}$                | Tangential derivative of $n_r^{\pm}$, $-(dn_r)^{\top}$ | Self-adjoint, yields curvatures                |
| $K_p^{n_r}$, $H_p^{n_r}$ | Det(A), Tr(A)/(n-1)                                    | $n_r$-Gauss–Kronecker and mean curvatures      |

This framework, rooted in the construction of $n_r^{\pm}$-Gauss maps, is essential for the analysis and classification of codimension-two bulk extremal surfaces in Lorentzian geometry and remains widely applicable in mathematical physics, particularly in geometric analysis and relativistic field theories.

Source: https://www.emergentmind.com/topics/codimension-two-bulk-extremal-surfaces