Background on Geometric Measure Theory | Gaoming Wang

Varifolds

Mon, 29 Jun 2026 00:00:00 +0000

Definition 3.1.1

An \(n\)-varifold \(V\) in \(\mathbb{R}^{n+k}\) is a Radon measure on \(\mathbb{R}^{n+k}\times G(n+k,n)\), where \(G(n+k,n)\) is the set of all \(n\)-dimensional subspaces in \(\mathbb{R}^{n+k}\).

Let \(M \subset \mathbb{R}^{n+k}\) be an \(n\)-dimensional manifold, and let \(\theta\) be an \(\mathcal{H}^n\)-measurable function on \(M\). Then the \(n\)-varifold \(V=|(M,\theta)|\) is defined by

\[ V(U)=\int_{(x,T_xM)\in U} \theta(x) d\mathcal{H}^n|_M(x). \]

Equivalently, for any continuous compactly supported function \(f\), we have

\[ V(f)=\int_{ } \theta(x)f(x,T_xM) d \mathcal{H}^n|_{M}(x). \]

Thus, a varifold arising from an \(n\)-dimensional manifold records tangent-plane information as well. If \(\theta=1\), we usually write \(|(M,1)|=|M|\). For example, \(P\) is the tangent plane of \(M\) at \(x\) if and only if the measure \(|M|\) restricted to \(\{ x \} \times G(n+k,n)\) is nonzero.

Definition 3.1.2

The weight measure \(\|V\|\) of an \(n\)-varifold \(V\) is defined by

\[ \|V\|(A)=V(A \times G(n+m,n)), \]

for any Borel subset \(A \subset \mathbb{R}^{n+m}\). Hence, \(\|V\|\) is a Radon measure on \(\mathbb{R}^{n+m}\). The support of \(\|V\|\), denoted by \(\mathrm{spt}\|V\|\), is defined by

\[ \mathrm{spt}\|V\|=\left\{ x \in \mathbb{R}^{n+k}: \|V\|(B^{n+k}_r(x))\gt{}0 \text{ for any }r\gt{}0 \right\}. \]

The (\(n\)-dimensional) density of \(\|V\|\) at \(x\) is defined by

\[ \Theta(\|V\|,x)=\lim_{r\to 0^+}\frac{\|V\|(B^{n+k}_r(x))}{\omega_nr^n}, \]

if the limit exists, where \(\omega_n\) is the volume of the unit ball in \(\mathbb{R}^n\).

Theorem 3.1.3

Suppose \(V_i\) is a sequence of \(n\)-varifolds such that for every compact \(K \subset \mathbb{R}^{n+k}\), there exists \(C=C(K)\) with

\[ V_i(K \times G(n+k,n))\leq C, \]

then, up to a subsequence, we can find \(V_i \to V\) in the sense of Radon measures. (The convergence is in the varifold sense.) Equivalently,

\[ \lim_{i \to \infty}V_i(f)=V(f) \]

for any \(f \in C_c(\mathbb{R}^{n+k}\times G(n+k,n))\).

Suppose \(V_n\) is defined by

\[ \sum_{i=1}^{2^n} |([0,1]\times \{ \frac{i}{2^n} \}, \frac{i}{2^n})|, \]

Then it converges to \(V\) in the varifold sense, where

\[ V(f)=\int_{0}^{1} \int_{0}^{1} f(x,y, \{ x_2=0 \})dx dy. \]

Note that \(\mathrm{spt}\|V\|=[0,1]^{2}\), so \(V\) cannot be written as \(V=|(M,\theta)|\) for any one-dimensional manifold \(M\).

Definition 3.1.4

We say \(M\) is countably \(n\)-rectifiable if \(M \subset N \cup \bigcup_{j=1}^\infty N_j\) where \(\mathcal{H}^n(N)=0\) and each \(N_j\) is an \(n\)-dimensional embedded \(C^1\) submanifold of \(\mathbb{R}^{n+k}\).

Equivalently, \(M\) is countably \(n\)-rectifiable if and only if there exists a countable family of Lipschitz maps \(f_j: \mathbb{R}^n \to \mathbb{R}^{n+k}\) such that

\[ M = N \cup \bigcup_{j=1}^\infty f_j(A_j), \]

where \(A_j \subset \mathbb{R}^n\) and \(\mathcal{H}^n(N)=0\).

For any countably \(n\)-rectifiable set \(M\), we write \(T_xM\) for the approximate tangent space of \(M\).

Definition 3.1.5

Let \(M\) be an \(\mathcal{H}^n\)-measurable subset of \(\mathbb{R}^{n+k}\) with \(\mathcal{H}^n(M\cap K)\lt{}+\infty\) for every compact subset \(K\). We say that an \(n\)-dimensional subspace \(P\) is an approximate tangent space of \(M\) at \(x\) if and only if

\[ \lim_{r\to 0^+} \int_{ \eta_{x,r}(M)}f(y) d\mathcal{H}^n(y) = \int_{P} f(y) d\mathcal{H}^n(y),\quad \text{for any }f \in C_c(\mathbb{R}^{n+k}). \]

Theorem 3.1.6

If \(M\) is rectifiable, then for \(\mathcal{H}^n\)-almost every \(x \in M\), there exists a unique approximate tangent space \(T_xM\) of \(M\) at \(x\).

Definition 3.1.7

We say an \(n\)-varifold \(V\) is rectifiable if there exists a countably \(n\)-rectifiable, \(\mathcal{H}^n\)-measurable subset \(M\) of \(\mathbb{R}^{n+k}\) and a positive locally \(\mathcal{H}^n\)-integrable function \(\theta\) on \(M\) such that

\[ V(f)=\int_{ M} f(x,T_xM)\theta(x)d \mathcal{H}^n(x) \]

We use the notation \(V=|(M,\theta)|\) for the varifold associated with \(M\) and \(\theta\).

Definition 3.1.8

Let \(V\) be an \(n\)-varifold in \(\mathbb{R}^{n+k}\) and let \(F: \mathbb{R}^{n+k} \to \mathbb{R}^{m+l}\) be a \(C^1\) map. The pushforward varifold \(F_{\#}V\) is defined by

\[ F_{\#}V(\phi) = \int_{ } \phi(F(x), DF_x(S)) \left\vert J_F(x, S) \right\vert \, dV(x,S), \]

for any continuous function \(\phi\) with compact support on \(\mathbb{R}^{m+l} \times G(m+l, n)\), where \(J_F(x, S)\) is the Jacobian of \(F\) restricted to \(S\), i.e.,

\[ J_F(x, S) = \sqrt{\det(\frac{\partial F}{\partial \tau_i} \cdot \frac{\partial F}{\partial \tau_j})}, \]

where \(\{ \tau_i \}_{i=1}^n\) is an orthonormal basis of \(S\).

Let \(F_t\) be a one-parameter family of diffeomorphisms on \(\mathbb{R}^{n+k}\) with \(F_0\) being the identity map. The first variation of an \(n\)-varifold \(V\) under the variation \(F_t\) is defined by

\[ \left.\frac{d}{dt}\right|_{t=0} \| (F_t)_{\#} V \|(K), \]

for any compact subset \(K \subset \mathbb{R}^{n+k}\).

Theorem 3.1.9

We have

\[ \left.\frac{d}{dt}\right|_{t=0} \| (F_t)_{\#} V \|(K) = \int_{ } \mathrm{div}^S \varphi(x) dV(x,S). \]

Definition 3.1.10

We define the first variation of \(V\) as a linear functional \(\delta V\) on \(C_c(\mathbb{R}^{n+k}, \mathbb{R}^{n+k})\) by

\[ \delta V(\varphi) = \int_{ } \mathrm{div}^S \varphi(x) dV(x,S). \]

Definition 3.1.11

We say that \(V\) has bounded first variation if \(\delta V\) is a bounded linear functional on \(C_c(\mathbb{R}^{n+k}, \mathbb{R}^{n+k})\).

\[ \delta V(\varphi) = \int \langle \nu_V(x), \varphi(x) \rangle d\|\delta V\|(x). \]

In particular, we can decompose \(\|\delta V\|=h\|V\|+\sigma_V\) into the absolutely continuous part \(h\|V\|\) and the singular part \(\sigma_V\) with respect to \(\|V\|\).

Let \(\mu\) and \(\nu\) be two Radon measures on \(\mathbb{R}^{n+k}\). We say \(\mu\) is absolutely continuous with respect to \(\nu\) (denoted \(\mu \ll \nu\)) if for every Borel set \(A\), \(\nu(A) = 0\) implies \(\mu(A) = 0\). By the Radon-Nikodym theorem, if \(\mu \ll \nu\), then there exists a \(\nu\)-measurable function \(h\) such that

\[ \mu(A) = \int_A h \, d\nu \]

for all Borel sets \(A\).

Conversely, a measure \(\mu\) is singular with respect to \(\nu\) if there exists a Borel set \(A\) such that \(\nu(A) = 0\) and \(\mu(\mathbb{R}^{n+k} \setminus A) = 0\). Intuitively, \(\mu\) and \(\nu\) are supported on disjoint sets.

Any Radon measure \(\mu\) can be uniquely decomposed as

\[ \mu = \mu_{\mathrm{ac}} + \mu_{\mathrm{sing}}, \]

where \(\mu_{\mathrm{ac}} \ll \nu\) and \(\mu_{\mathrm{sing}} \perp \nu\). So, if \(V\) has bounded first variation, we can write

\[ \delta V(\varphi) =-\int \langle \vec{H}, \varphi(x) \rangle d\|V\|(x) + \int \langle \nu_V(x), \varphi(x) \rangle d\sigma_V(x). \]

where \(\vec{H} =-h \nu_V\) is called the generalized mean curvature vector of \(V\), and \(\nu_V\sigma_V\) is called the generalized boundary of \(V\).

Suppose \(M=\{ (\cos \theta, \sin \theta): \theta \in (0,\pi) \}\), the 1-dimensional half-circle in \(\mathbb{R}^2\), and \(V=|M|\) is the associated varifold. Then, we have

\[ \begin{aligned} \delta V(\varphi)={}&\int_{ M} \mathrm{div}^M \varphi(x) d\mathcal{H}^1|_M(x)=\int_{ M} \mathrm{div}^M \varphi^\top(x) d\mathcal{H}^1|_M(x)+\int_{ M} \varphi(x) \cdot x\\ ={}& \varphi(1,0) \cdot (0,-1) + \varphi(-1,0) \cdot (0,-1) + \int_{ M} \varphi(x) \cdot x d\mathcal{H}^1|_M(x). \end{aligned} \]

So the generalized mean curvature vector \(\vec{H}=-x\) is just the usual mean curvature vector of \(M\), and the generalized boundary \(\nu_V\sigma_V=(0,-1)\delta_{(1,0)} + (0,-1)\delta_{(-1,0)}\) represents the two boundary points of \(M\) with the corresponding outward normal vectors.

Suppose \(M=[0,1]\times \{ 0 \}, \theta(x,y)=x\), and \(V=|(M,\theta)|\) is the associated varifold. Then, we have

\[ \delta V = \int_{ 0}^1 \frac{\partial }{\partial x} \varphi^\top(x,0) x dx = \varphi^\top(1,0)\cdot (1,0) - \int_{ 0}^1 \varphi^\top(x,0)\cdot (1,0) dx. \]

So we have

\[ \vec{H}=(\frac{1}{x},0),\quad \nu_V\sigma_V=(1,0)\delta_{(1,0)}. \]

Thus, the generalized mean curvature vector \(\vec{H}\) depends not only on the geometry of \(M\), but also on the weight function \(\theta\).

This example also shows that, unlike the case of smooth submanifolds, the generalized mean curvature vector of a varifold may not be perpendicular to the tangent plane. If we restrict to integral rectifiable varifolds, we have the following result of Brakke [Bra15].

Theorem 3.1.12

Suppose \(V\) is an integral rectifiable varifold with bounded first variation, then the generalized mean curvature vector \(\vec{H}\) is perpendicular to the tangent plane of \(V\) for \(\|V\|\)-almost every point.

Definition 3.1.13

We say that \(V\) is a stationary varifold in \(U\) if for any \(\varphi \in C_c(U,\mathbb{R}^{n+k})\), we have

\[ \int_{ } \mathrm{div}^S \varphi(x)dV(x,S)=0. \]

If \(V=|M|\), then \(V\) being stationary means

\[ \int_{ M} \mathrm{div}^M \varphi(x) d\mathcal{H}^n|_M(x)=0. \]

This is equivalent to saying that \(M\) is minimal.

This notion of stationarity is very weak. Triple-junctions of three half-planes meeting at \(120\) degrees are stationary.

Theorem 3.1.14

Let \(V\) be a stationary \(n\)-varifold in \(\mathbb{R}^{n+k}\). Then the function

\[ \Phi(r) = \frac{\|V\|(B_r(x_0))}{\omega_n r^n} \]

is monotone increasing in \(r \gt{} 0\). Moreover, we have

\[ \Phi(r)-\Phi(s)=\int_{ B_r(x_0)\backslash B_s(x_0)} \frac{|(y-x_0)^\perp|^2}{|y-x_0|^{n+2}} d\|V\|(y) \]

Proof. We can choose test vector fields \(\varphi\) as before. Suppose \(x_0=0\) for simplicity.

\[ \varphi = \begin{cases} y\left( \frac{1}{|y|^n}-\frac{1}{\rho^n} \right), & \sigma\leq |y|\leq \rho\\ y\left( \frac{1}{\sigma^n}-\frac{1}{\rho^n} \right), & |y|\lt{}\sigma\\ \end{cases} \]

We insert \(\varphi\) into the first variation formula and get the desired monotonicity formula. ◻

One can obtain a modified monotonicity formula for varifolds with \(L^p\)-integrable mean curvature (\(p\gt{}n\)) and no generalized boundary.

Theorem 3.1.15

Suppose \(V_i\) is a sequence of rectifiable stationary varifolds in \(U\) and for any \(K \subset \subset U\), \(\sup\|V_i\|(K)\lt{}+\infty\). We also assume \(\Theta(\|V_i\|,x)\geq 1\) for almost every \(x\in \mathrm{spt}\|V_i\|\). Then, up to a subsequence, there exists a rectifiable stationary varifold \(V\) in \(U\) such that \(V_i\to V\) in the varifold sense.

In particular, if each \(V_i\) is integral, so is \(V\).

The condition \(\Theta\geq 1\) is essential.
Stationarity is also essential: \(V_n=\sum_{i=1}^{2^n}|[\frac{2i-1}{2^{n+1}}, \frac{2i}{2^{n+1}}]|\) converges to \(|([0,1],\frac{1}{2})|\).

Definition 3.1.16

The tangent cone of an \(n\)-varifold \(V\) at \(x\), denoted by \(\mathrm{VarTan}(V,x)\), is defined by

\[ \mathrm{VarTan}(V,x):= \{V':V'=\lim_{i\to \infty}(\eta_{x,\rho_i})_\#V\text{ for some }\rho_i\to 0^+ \}. \]

The tangent cone at infinity, denoted by \(\operatorname{VarTan}(V,\infty)\), is defined by

\[ \mathrm{VarTan}(V,\infty):= \{V':V'=\lim_{i\to \infty}(\eta_{0,\rho_i})_\#V\text{ for some }\rho_i\to +\infty\}. \]

The tangent cone of a varifold may not be unique. Right now, we do not even know if the tangent cone of any stationary rectifiable varifold is unique or not. This is still an open problem.

Proposition 3.1.17

If \(V\) is stationary in \(\mathbb{R}^{n+k}\), then the tangent cone of \(V\) at \(\infty\) or \(x\) is a stationary cone.

Finally, for any \(n\)-varifold defined on \(U\), \(\mathrm{reg}\|V\|\) denotes the regular set of \(\mathrm{spt}\|V\|\) in \(U\), i.e., the set of points \(x \in \mathrm{spt}\|V\|\) such that there exists \(r\gt{}0\) with \(B^{n+1}_r(x)\subset U\) and \(\mathrm{spt}\|V\|\cap B^{n+1}_r(x)\) is a smooth (immersed) hypersurface in \(B^{n+1}_r(x)\). \(\mathrm{sing}\|V\|\) denotes the singular set of \(\mathrm{spt}\|V\|\) in \(U\).

Allard’s regularity theorem is a foundational result in geometric measure theory. It gives conditions under which a stationary varifold is regular (i.e., smooth) near a point.

Theorem 3.1.18

Let \(V\) be an \(n\)-dimensional stationary integral rectifiable varifold in \(B_2(0)\subset \mathbb{R}^{n+k}\), and suppose that

\[ \|V\|(B_2(0))\leq (1+\delta)\omega_n 2^n \]

for some \(0\lt{}\delta\lt{}1\). Then, there exists \(\varepsilon=\varepsilon(n,\delta)\) such that if

\[ E^{2}:=\int_{ B_2(0)} \operatorname{dist}^2(x,P)\,d\|V\|(x)\leq \varepsilon \]

where \(P=\{ x_{n+1}=0 \}\), then there exists a function \(u \in C^{1,\alpha}(\bar{B_1^n(0)})\) such that

\[ \mathrm{spt}\|V\| \cap B_1(0) = \{ (x',u(x')): x'\in B_1^n(0) \} \]

and

\[ \sup_{B_1^n(0)} |u| + \|D u\|_{L^{\infty}(B_1^n(0))} + [D u]_{C^{0,\alpha}(B_1^n(0))} \leq C(n,\delta) E. \]

Note that \(\delta\) cannot be 1 since otherwise, we have the scaled catenoid as a counterexample.

Theorem 3.1.19

Let \(V\) be an \(n\)-dimensional stationary integral rectifiable varifold in an open set \(U \subset \mathbb{R}^{n+k}\). Suppose that the density of \(\|V\|\) at a point \(x_0 \in U\) is \(1\). Then \(x_0\in \mathrm{reg}\|V\|\).

Sets of Finite Perimeter

Mon, 29 Jun 2026 00:00:00 +0000

We collect some basic definitions and facts about the set of finite perimeter.

If \(E\) is a set with smooth boundary, then we have the following Gauss-Green formula:

\[ \int_{E} \mathrm{div} \varphi d\mathcal{H}^n=\int_{ \partial E} \varphi \cdot \nu d\mathcal{H}^{n-1}, \]

for any \(\varphi \in C_c^1(\mathbb{R}^n,\mathbb{R}^n)\). In particular, if we require that \(|\varphi|\leq 1\), then the right-hand side can be bounded by the perimeter of \(E\) as

\[ \int_{ \partial E} \varphi\cdot \nu d\mathcal{H}^{n-1}\leq \mathcal{H}^{n-1}(\partial E). \]

and equality holds if and only if \(\varphi=\nu\) on \(\partial E\). This fact motivates the following definition of the perimeter of \(E\) as

\[ |\partial E|= \sup_{\varphi \in C_c^1(\mathbb{R}^n,\mathbb{R}^n), |\varphi|\leq 1} \int_{ \partial E} \varphi\cdot \nu d\mathcal{H}^{n-1}. \]

Note that the right-hand side of the equation above does not depend on the regularity of the boundary of \(E\). This motivates the following definition.

Definition 3.2.1

The perimeter of a set \(E\) in an open set \(U\) is defined as

\[ P(E,U)= \sup_{\varphi \in C_c^1(U,\mathbb{R}^n), |\varphi|\leq 1} \int_{ U\cap E} \mathrm{div} \varphi d\mathcal{H}^n. \]

We say that \(E\) has locally finite perimeter in \(U\) if \(P(E,W)\lt{}+\infty\) for any \(W \subset \subset U\). Such a set is also called a Caccioppoli set.

Suppose \(E\) has locally finite perimeter in \(U\). Then, we can consider the linear functional \(J_E\) on \(C_c^1(U,\mathbb{R}^n)\) defined by

\[ J_E(\varphi)= \int_{ U\cap E} \mathrm{div} \varphi d\mathcal{H}^n. \]

This functional is clearly linear. Since we have assumed that \(E\) has locally finite perimeter, we have that \(J_E\) is a bounded linear functional on \(C_c^1(W,\mathbb{R}^n)\) for any \(W \subset \subset U\). (Recall that an operator \(T\) is bounded if \(\sup_{|\varphi|\leq 1} |T(\varphi)|\lt{}+\infty\).) Now, we can apply the Riesz Representation Theorem to get a unique Radon measure \(\mu_E\) on \(U\), and a vector-valued function \(\nu_E\) on \(U\) such that \(|\nu_E(x)|=1\) for \(\mu_E\)-almost every \(x\) and

\[ J_E(\varphi)= \int_{ U} \varphi\cdot \nu_E d\mu_E, \]

for any \(\varphi \in C_c^1(U,\mathbb{R}^n)\).

The vector-valued measure \(\nu_E \mu_E\) is called the Gauss-Green measure of \(E\) in \(U\). We use \(\overrightarrow{\mu}_E\) to denote the vector-valued measure \(\nu_E \mu_E\). Then, the perimeter of \(E\) in \(U\) can also be written as

\[ P(E,U)= \mu_E(U). \]

We can understand \(\mu_E\) as a boundary measure of \(E\) in \(U\), and \(\nu_E\) as a boundary normal vector of \(E\) in \(U\), pointing outward.

Suppose \(E=[0,+\infty)^{2} \in \mathbb{R}^2\). Then, \(E\) is a set with locally finite perimeter. In particular,

\[ \mu_E=\mathcal{H}^1|_{[0,+\infty)\times \{0\}} + \mathcal{H}^1|_{\{0\}\times [0,+\infty)},\quad \nu_E=(0,-1)|_{[0,+\infty)\times \{0\}} + (1,0)|_{\{0\}\times [0,+\infty)}. \]

Recall that \(\mathrm{spt}\,\mu_E\) is the support of \(\mu_E\), which is defined as the set of points \(x\) such that \(\mu_E(B_r(x))\gt{}0\) for any \(r\gt{}0\).

Then, we have the following proposition:

Proposition 3.2.2

Suppose \(E\) is a set of locally finite perimeter in \(\mathbb{R}^n\). Then, \(\mathrm{spt}\,\mu_E\subset \partial E\), where \(\partial E\) is the topological boundary of \(E\).

Let

\[ E := (0,1)^2 \backslash \{ 0.5 \}\times [0,0.5]\subset \mathbb{R}^2. \]

Then \(\partial E\) contains both the outer boundary of the square and the slit:

\[ \partial E=\partial(0,1)^2\;\cup\;\bigl(\{0.5\}\times[0,0.5]\bigr). \]

On the other hand, removing a \(1\)-dimensional set does not change \(\chi_E\) in \(L^1\), so the perimeter measure is the same as for the open square:

\[ \mu_E=\mu_{(0,1)^2},\qquad \mathrm{spt}\,\mu_E=\partial(0,1)^2. \]

Hence

\[ \mathrm{spt}\,\mu_E \subsetneq \partial E. \]

To better illustrate the “true” boundary of a set of finite perimeter in the measure sense, we introduce the reduced boundary.

Definition 3.2.3

Given a set \(E\) of locally finite perimeter in \(\mathbb{R}^n\), the reduced boundary \(\partial^*E\) is the set of points \(x\in\mathbb{R}^n\) such that the limit

\[ \nu_E(x):=\lim_{r \to 0^+} \frac{\overrightarrow{\mu}_E(B_r(x))}{\mu_E(B_r(x))} \]

exists and belongs to \(\mathbb{S}^{n-1}\).

For the \(E=(0,1)^2 \backslash \{ 0.5 \}\times [0,0.5]\), we have

\[ \partial^* E = \partial(0,1)^2\backslash \{ (0,0), (1,0), (0,1), (1,1) \}. \]

In particular, we have

\[ \lim_{r \to 0^+} \frac{\overrightarrow{\mu}_E(B_r(0))}{\mu_E(B_r(0))} = (-\frac{1}{2},-\frac{1}{2})\notin \mathbb{S}^{n-1}. \]

So \((0,0)\notin \partial^* E\).

We have the following structure theorem for the reduced boundary.

Theorem 3.2.4

Suppose \(E\) is a set of locally finite perimeter in \(\mathbb{R}^n\). Then, the reduced boundary \(\partial^* E\) is a \((n-1)\)-countably rectifiable set in \(\mathbb{R}^n\), and we actually have

\[ \mu_E = \mathcal{H}^{n-1}|_{\partial^* E},\quad \nu_E = \text{ the outer normal vector field of } \partial^* E. \]

Note that since \(\partial^* E\) is rectifiable, so the approximate tangent space \(T_x \partial^* E\) is well-defined for \(\mu_E\)-almost every \(x \in \partial^* E\). Hence, we can choose the outer normal vector field perpendicular to \(T_x \partial^* E\) and point outward of \(E\).

Definition 3.2.5

Given Lebesgue measurable sets \(\{E_h\}_{h\in\mathbb{N}}\) and \(E\) in \(\mathbb{R}^n\), we say that \(E_h\) locally converges to \(E\), and write \(E_h \xrightarrow{\mathrm{loc}} E\), if

\[ \lim_{h \to \infty} \left| K \cap (E \Delta E_h) \right| = 0, \qquad \forall K \subset \mathbb{R}^n \text{ compact.} \]

This is equivalent to say that \(\chi_{E_h} \xrightarrow{} \chi_{E}\) in \(L^1_{loc}(\mathbb{R}^n)\).

Proposition 3.2.6

If \(\{E_h\}_{h\in\mathbb{N}}\) is a sequence of sets of locally finite perimeter in \(\mathbb{R}^n\), with

\[ E_h \xrightarrow{\mathrm{loc}} E, \qquad \limsup_{h \to \infty} P(E_h; K) \lt{} \infty, \]

for every compact set \(K\) in \(\mathbb{R}^n\), then \(E\) is of locally finite perimeter in \(\mathbb{R}^n\), \(\mu_{E_h} \rightharpoonup^* \mu_E\) and, for every open set \(A \subset \mathbb{R}^n\), we have

\[ P(E;A) \leq \liminf_{h \to \infty} P(E_h;A). \]

The inequality above can be strict.

Let \(B\) denote the closed unit disc \(B = \{x \in \mathbb{R}^2 : |x| \leq 1\}\). For each \(i \in \mathbb{N}\), let \(E_i = B \setminus \bigcup_{k=1}^{n_i} \bar{B}_{r_i}(x_{i,k})\), where \(\{x_{i,k}\}_{k=1}^{n_i}\) is a collection of centers inside \(B\), \(n_i\) increases as \(i \to \infty\), and \(r_i\to 0\) in such a way that \(n_i r_i\to c\gt{}0\) and

\[ \bigcup_{k=1}^{n_i} \bar{B}_{r_i}(x_{i,k}) \subset B, \]

with all the small discs disjoint and contained in \(B\).

For each \(E_i\), the perimeter is

\[ P(E_i) = P(B) + n_i P(\bar{B}_{r_i}) = 2\pi + n_i \cdot 2\pi r_i, \]

since each removed disc adds \(2\pi r_i\) to the perimeter.

As \(i \to \infty\), the number of holes \(n_i \to \infty\) and the radii \(r_i \to 0\), so the total removed area goes to \(0\). In the limit, the set \(E_i\) converges (locally in measure, or in \(L^1_{\mathrm{loc}}\)) to \(B\). However,

\[ \lim_{i \to \infty} P(E_i) = 2\pi+2\pi c, \]

because the total boundary length of the small holes remains visible before passing to the limit.

Thus, the perimeter functional is lower semicontinuous under local convergence, and it is possible that

\[ P(\lim_{i\to\infty} E_i) \lt{} \liminf_{i\to\infty} P(E_i), \]

because the limiting set does not retain the interior boundaries present in the approximating sequence.

Let \(D = [0,4] \times [0,2] \subset \mathbb{R}^2\). For each \(i\in\mathbb{N}\), construct \(E_i\) as the subset of \(D\) with lower and side boundaries of \(D\), but the top replaced with a zig-zag curve of \(n_i\) “teeth,” each of amplitude \(h_i\) (height above \(y=2\)), so that as \(i\to\infty\), \(n_i\to\infty\) and \(h_i\to 0\).

Each \(E_i\) is the set (shaded in the pictures above) contained below its zig-zag boundary and above \(y=0\). As \(i\to\infty\), the upper boundary of \(E_i\) converges to the straight line \(y=2\), and \(E_i\) converges (in measure) to \(D\).

The perimeter of \(E_i\) is

\[ P(E_i) = 2\cdot 2 + 4 + L_{\text{zigzag}}, \]

where \(L_{\text{zigzag}}\) is the total length of the zig-zag curve. If each tooth projects horizontally \(\delta x = 4/n_i\) and has vertical height \(h_i\), then

\[ L_{\text{zigzag}} = n_i \cdot 2\sqrt{ \left(\frac{\delta x}{2}\right)^2 + h_i^2 } = n_i \cdot 2 \sqrt{ (2/n_i)^2 + h_i^2 }. \]

If \(n_i h_i\to a\gt{}0\), then the zig-zag boundary length satisfies

\[ \lim_{i\to\infty} L_{\text{zigzag}} = 2\sqrt{4+a^2}\gt{}4. \]

Thus, in the limit, the region \(E_i\) converges to \(D\), but the perimeter keeps a positive excess:

\[ P(\lim_{i\to\infty} E_i) \lt{} \liminf_{i\to\infty} P(E_i). \]

This example again shows lower semicontinuity: the limiting domain loses the additional oscillating boundary length in the limit.

Theorem 3.2.7

If \(R\gt{}0\) and \((E_k)_{k\in\mathbb{N}}\) are sets of finite perimeter in \(\mathbb{R}^n\), with

\[ \begin{aligned} \sup_{k\in\mathbb{N}} P(E_k) &\lt{} \infty,\\ E_k &\subset B_R,\qquad \forall\, k\in\mathbb{N}, \end{aligned} \]

then one can find a set \(E\) of finite perimeter in \(\mathbb{R}^n\) and a subsequence of \((E_k)_{k\in\mathbb{N}}\), still denoted by \(E_k\), such that

\[ \chi_{E_{k}} \to \chi_E,\qquad \mu_{E_{k}} \stackrel{*}{\rightharpoonup} \mu_E,\qquad E \subset B_R. \]

Definition 3.2.8

Suppose \(A\subset \mathbb{R}^n\) is a bounded set and \(E_0\) is a set of finite perimeter in \(\mathbb{R}^n\). We say \(E_0\) is a perimeter minimizer in \(A\) if

\[ P(E_0;A)\leq P(E;A) \]

for every set of finite perimeter \(E\) such that \(E\setminus A=E_0\setminus A\).

In particular, we say \(E_0\) is a perimeter minimizer in \(\mathbb{R}^n\) if \(E_0\) is a perimeter minimizer for any bounded set \(A\).

Proposition 3.2.9

Suppose \(\{ E_k \}\) is a sequence of perimeter minimizers in \(\mathbb{R}^n\), and assume \(E_k\) converges to \(E\) in \(L^1_{\mathrm{loc}}(\mathbb{R}^n)\). Then \(E\) is also a perimeter minimizer in \(\mathbb{R}^n\).

Now, we are ready to state the general existence of minimizers in the following sense.

Proposition 3.2.10

Let \(A \subset \mathbb{R}^n\) be a bounded set and let \(E_0\) be a set of finite perimeter in \(\mathbb{R}^n\). Then there exists a set of finite perimeter \(E\) such that \(E \setminus A = E_0 \setminus A\) and

\[ P(E) \leq P(F) \]

for every \(F\) such that \(F \setminus A = E_0 \setminus A\).

For example, given a boundary curve \(\Gamma\), which lies on a boundary of a convex domain \(D\). We can construct a set of finite perimeter \(E\) such that \(\partial(\partial E \cap \Omega) = \Gamma\) and it is the one with the minimal perimeter.

The key here is actually the regularity of the boundary of the set of finite perimeter.

Dimension Reduction and Regularity of Perimeter Minimizers

Mon, 29 Jun 2026 00:00:00 +0000

Theorem 3.3.1

Suppose \(E\) is a perimeter minimizer in \(\mathbb{R}^{n+1}\). We denote \(V=|\partial^* E|\). Then, we have the following regularity result: \(\mathrm{sing}\|V\|=\emptyset\) if \(n\leq 6\), \(\mathrm{sing}\|V\|\) is discrete if \(n=7\), and \(\mathcal{H}^{n-7+\delta}(\mathrm{sing}\|V\|)=0\) for any \(\delta\gt{}0\) for \(n\geq 8\).

Theorem 3.3.2

Let \(\mathcal{V}\) be the collection of all the varifolds in the preceding theorem, i.e., the varifolds corresponding to the perimeter minimizers. Then,

\[ \operatorname{dim}(\mathrm{sing}(\|V\|\cap B_1))\leq n-7. \]

In particular, if \(n=7\), then \(\mathrm{sing}(\|V\|\cap B_1)\) is discrete.

Proof. We denote \(F^l=\{ V \in \mathcal{V}: \mathcal{H}^l(\mathrm{sing}\cap B_1)\gt{}0 \}\).

Proposition 3.3.3

For each \(V \in F^l\), there exists \(\boldsymbol{C} \in \mathrm{VarTan}(V,x)\cap F^l\) for \(\mathcal{H}^l-\)a.e. \(x \in \mathrm{sing}(\|V\|)\cap B_1\).

Proof. Recall that we actually have for \(\mathcal{H}^l\)-a.e. \(x \in \mathrm{sing}(\|V\|)\cap B_1\), we have

\[ \limsup_{r \to 0} \frac{\mathcal{H}^l_\infty(\text{sing}\|V\|\cap B_r(x))}{\omega_n r^l}\gt{}0. \]

This is equivalent to \(\mathcal{H}^l(A)\gt{}0 \equiv \mathcal{H}^l_\infty(A)\gt{}0\) for any \(A \subset \mathbb{R}^n\).

We choose \(r_i \to 0\) such that

\[ \lim_{i \to \infty} \frac{\mathcal{H}^l_\infty(\text{sing}\|V\|\cap B_{r_i}(x))}{\omega_n r_i^l}\gt{}0. \]

By taking a subsequence, we can assume \((\eta_{x,r_i})_{\#}V\) converges to \(\boldsymbol{C} \in \mathrm{VarTan}(V,x)\).

If \(\mathcal{H}^l_\infty(\mathrm{sing}\,\|\boldsymbol{C}\|)=0\), then for any \(\varepsilon\gt{}0\), we can find a covering of \(\mathrm{sing}\,\|\boldsymbol{C}\|\) by balls \(\{ B_{s_j}(y_j) \}_{j=1}^\infty\) such that

\[ \sum_{j=1}^\infty s_j^l \lt{} \varepsilon. \]

Note that \(\mathrm{sing}\,\|\boldsymbol{C}\|\cap B_1(0)\) is compact, we know \(\mathrm{sing}\,\|(\eta_{x,r_i})_{\#}V\|\cap B_1(0)\) can also be covered by \(\{ B_{s_j}(y_j) \}_{j=1}^\infty\) for \(i\) large enough. Thus, we have

\[ \frac{\mathcal{H}^l_\infty(\mathrm{sing}\,\|V\|\cap B_{r_i}(x))}{r_i^l}\lt{}\varepsilon \]

for \(i\) large enough, which is a contradiction. ◻

Now we apply the above proposition iteratively to obtain a sequence of varifolds \(\{ V_k \}_{k=0}^{K}\) such that:

\(V_0=V\).
\(V_{k+1} \in \mathrm{VarTan}(V_k,x_k)\) for some \(x_k \in \mathrm{sing}(\|V_k\|)\cap B_1 \backslash \mathcal{S}(V_k)\).
\(\operatorname{dim}(\mathcal{S}(V_{k+1}))\gt{}\operatorname{dim}(\mathcal{S}(V_k))\) for each \(0\leq k \leq K-1\).
\(\mathcal{H}^l(\mathrm{sing}(\|V_k\|)\cap B_1)\gt{}0\) for each \(0\leq k \leq K\).
\(V_K=\boldsymbol{C}\times \mathbb{R}^{m}\) where \(\boldsymbol{C}\backslash \{0\}\) is a smooth immersed cone for some \(m\geq 0\).

In particular, the last two conditions imply \(m=l\). Now, \(\boldsymbol{C}\) becomes a stable cone with isolated singular point \(0\) of dimension \(n-l\). By the classification of the stable cones, we know \(n-l\geq 7\), and hence \(l\leq n-7\). This shows \(\operatorname{dim}(\mathrm{sing}\|V\|\cap B_1)\leq n-7\).

In the case \(n=7\), we have

\[ \mathcal{H}^\alpha(\mathrm{sing}\,\|V_K\|\cap B_1)=0 \]

for any \(\alpha\gt{}0\) by the above result.

If \(\mathrm{sing}(\|V\|\cap B_1)\) is not discrete, then we can find \(x_j \in \mathrm{sing}(\|V\|\cap B_1)\) such that \(x_j \to x_0 \in \mathrm{sing}(\|V\|\cap B_1)\). Now, up to a subsequence, we can assume \((\eta_{x_0,|x_j-x_0|})_{\#}V\) converges to \(\boldsymbol{C} \in \mathrm{VarTan}(V,x_0)\) and we denote \(\xi = \lim_{j \to \infty} \frac{x_j-x_0}{|x_j-x_0|} \neq 0\). So \(\mathcal{S}(\boldsymbol{C})\) contains the line spanned by \(\xi\). In particular, \(\mathcal{H}^1(\mathrm{sing}(\|\boldsymbol{C}\|)\cap B_1)\gt{}0\) which is a contradiction. Hence, \(\mathrm{sing}(\|V\|\cap B_1)\) is discrete. ◻

Proof of Bernstein Theorem

Mon, 29 Jun 2026 00:00:00 +0000

Now, we are ready to prove the Bernstein Theorem up to the dimension \(n\leq 7\).

Theorem 3.4.1

Suppose \(u\) is a solution of the minimal surface equation on \(\mathbb{R}^n\) and \(n\leq 7\). Then, \(u\) is an affine function.

More analysis on the stationary cones

Recall that the Jacobi operator is defined as

\[ L(\varphi) = \Delta \varphi + |A|^{2}\varphi \]

for any smooth function \(\varphi\) on the regular part of \(M\). If \(M^n\subset\mathbb{R}^{n+1}\) is a stationary cone with isolated singular point \(0\), then we can rewrite the Jacobi operator as

\[ L(\varphi)=\frac{\partial ^{2}\varphi}{\partial r^2}+\frac{n-1}{r}\frac{\partial \varphi}{\partial r}+\frac{1}{r^{2}}\left( \Delta^\Sigma \varphi + |A_\Sigma|^{2}\varphi \right) \]

where \(\Sigma=M \cap \mathbb{S}^{n}\). This is called the link of the stationary cone, which is a minimal hypersurface in \(\mathbb{S}^{n}\).

So it is important to study the operator \(\Delta^\Sigma + |A_\Sigma|^{2}\) on the link \(\Sigma\). Suppose \(\Sigma\) is not totally geodesic, we define \(\lambda_1(\Sigma)\) to be the first eigenvalue of the operator \(-\Delta^\Sigma - |A_\Sigma|^{2}\) by

\[ \lambda_1(\Sigma)=\inf_{\varphi \in H^1(\Sigma), \varphi \neq 0} \frac{\int_\Sigma |\nabla \varphi|^2 - |A_\Sigma|^2 \varphi^2 d\Sigma}{\int_\Sigma \varphi^2 d\Sigma} \]

Theorem 3.4.2

We have \(\lambda_1(\Sigma)\leq -(n-1)\).

Proof. We use the following Simons inequality for the link \(\Sigma\):

\[ |A_\Sigma|\Delta^\Sigma|A_\Sigma|+|A_\Sigma|^4\geq \frac{2}{n-1}|\nabla |A_\Sigma||^2+(n-1)|A_\Sigma|^2. \]

Now, we directly integrate the inequality over \(\Sigma\) to get

\[ \int_{\Sigma} |\nabla |A_\Sigma||^2 - |A_\Sigma|^4 \leq -(n-1)\int_{\Sigma} |A_\Sigma|^2. \]

Using \(|A_\Sigma|\) as a test function in the Rayleigh quotient gives

\[ \lambda_1(\Sigma)\leq \frac{\int_\Sigma |\nabla |A_\Sigma||^2-|A_\Sigma|^4\,d\Sigma} {\int_\Sigma |A_\Sigma|^2\,d\Sigma} \leq -(n-1). \]

◻

Proposition 3.4.3

Suppose \(\varphi\) is a smooth function on \(\Sigma\) achieving the infimum in the definition of \(\lambda_1(\Sigma)\). If \(u\) is a positive Jacobi field on \(M\), then if we define

\[ V(r):=\int_{ \Sigma} \varphi(x) u(r x) d\Sigma(x) \]

Then

\[ (V(r)r^{\kappa_n})'\leq 0 \]

for \(\kappa_n:=\frac{n-2}{2}-\sqrt{\frac{(n-2)^{2}}{4}-(n-1)}\).

Proof. We compute

\[ \begin{aligned} V''(r) &= \int_{\Sigma} \varphi(x) \frac{\partial^2}{\partial r^2}u(r x) \, d\Sigma(x) \\ &= -\frac{n-1}{r} V'(r) - \frac{1}{r^2} \int_{\Sigma} \varphi(x) \big( \Delta^\Sigma u(r x) + |A_\Sigma|^2 u(r x) \big) \, d\Sigma(x) \\ &= -\frac{n-1}{r} V'(r) - \frac{1}{r^2} \int_{\Sigma} u(r x) \big( \Delta^\Sigma \varphi(x) + |A_\Sigma|^2 \varphi(x) \big) \, d\Sigma(x) \\ &\leq -\frac{n-1}{r} V'(r) - \frac{n-1}{r^2} V(r) \end{aligned} \]

where in the last step we used that \(\varphi\) is an eigenfunction of the operator \(-\Delta^\Sigma - |A_\Sigma|^2\) with eigenvalue \(\lambda_1(\Sigma) \leq -(n-1)\), and that \(u\gt{}0\). The proof is finished by analyzing the resulting ODE inequality. ◻

We consider \(W(t)=V(t^{-1/\beta})t^{\gamma/\beta}\).

Then, we can choose \(\beta\) and \(\gamma\) such that \(W''(t)\leq 0\). To see this, we compute

\[ \begin{aligned} W''(t) &= \frac{1}{\beta^2} t^{\frac{\gamma}{\beta}-2} \left( V''(t^{-1/\beta})t^{-2/\beta} + V'(t^{-1/\beta})t^{-1/\beta}((1+\beta)-2\gamma) + V(t^{-1/\beta})\gamma(\gamma-\beta) \right) \\ \end{aligned} \]

So we have

\[ 1+\beta-2\gamma = \gamma(\gamma-\beta)=n-1 \]

The choice of \(\gamma = -\kappa_n\) and \(\beta = 2\sqrt{\frac{(n-2)^{2}}{4}-(n-1)}\) satisfies the above equations, and hence \(W''(t)\leq 0\). Since \(t=r^{-\beta}\) and \(W(t)=V(r)r^{\kappa_n}\), this concavity gives the desired monotonicity of \(V(r)r^{\kappa_n}\) in the \(r\) variable.

We are ready to prove the Bernstein Theorem. We need to study the blow-down limit of the graph of \(u\). This is due to De Giorgi [DG65].

Theorem 3.4.4

Let \(u\) solve the minimal surface equation on \(\mathbb{R}^n\) for \(n\leq 7\). Then the blow-down limit of the graph of \(u\) is a density-one hyperplane in \(\mathbb{R}^{n+1}\).

Proof. Suppose \(M\) is the graph of \(u\). Then it is a minimal hypersurface in \(\mathbb{R}^{n+1}\), and it is minimizing area in \(\mathbb{R}^{n+1}\).

Let \(P\) be the region \(\{ (x,y) \in \mathbb{R}^n \times \mathbb{R}: y\lt{}u(x) \}\). By the previous result, we know \(P\) is perimeter minimizer in \(\mathbb{R}^{n+1}\). We consider the blow-down limit of \(P\) defined as follows. Define \(P_r:=\frac{1}{r}P\). Up to a subsequence, \(P_r\) converges to a set \(P_\infty\) in \(L^1_{\mathrm{loc}}(\mathbb{R}^{n+1})\). By compactness, \(P_\infty\) is a perimeter minimizer in \(\mathbb{R}^{n+1}\). The boundary of \(P_r\), denoted by \(M_r\), converges to the boundary of \(P_\infty\) in the varifold sense; denote the limit by \(V_\infty\).

By the previous regularity result, we know \(\mathrm{sing}\,\|V_\infty\|\) is empty or discrete. If it is empty, then \(V_\infty\) is a density-one hyperplane in \(\mathbb{R}^{n+1}\). If it is discrete, then \(V_\infty\) can only have an isolated singular point at the vertex \(0\), since it is a stationary cone.

Now, we consider \(P'_r:=P_r+e_{n+1}\). This is again a perimeter minimizer in \(\mathbb{R}^{n+1}\), and we have \(P_r \subset P'_r\). Up to a subsequence, \(P'_r\) converges to a set \(P'_\infty\), and \(P'_\infty = P_\infty + e_{n+1}\). In particular, \(P_\infty \subset P'_\infty\). By the strong maximum principle, we have either \(P_\infty = P'_\infty\), or their boundaries are disjoint. In the first case, we are done, since \(V_\infty\) is translation invariant along \(e_{n+1}\) direction, it can only be a density-one hyperplane in \(\mathbb{R}^{n+1}\). In the second case, we know that \(V_\infty+\lambda e_{n+1}\) is disjoint from \(V_\infty\) for any \(\lambda\gt{}0\). Hence, we can construct a positive Jacobi field \(u\) on \(M_\infty\) by \(u:=\left< \nu_{M_\infty}, e_{n+1} \right>\) where \(M_\infty:=\mathrm{reg}\|V_\infty\|\). Now, for the function \(V(r):=\int_{ \Sigma} \varphi(x) u(r x) d\Sigma(x)\), we have

\[ V(r)\leq \int_{ \Sigma} \varphi(x) \leq C \]

where \(C\) is independent of \(r\). On the other hand, we have \(\kappa_7=2\), and hence

\[ V(r)\geq \frac{V(1)}{r^2} \text{ for } 0\lt{}r\lt{}1, \]

which implies \(V(r)\to +\infty\) as \(r\to 0\). This contradiction implies \(\Sigma\) is totally geodesic, and hence \(M_\infty\) is a density-one hyperplane in \(\mathbb{R}^{n+1}\). ◻

Proof. By Allard’s regularity theorem, \(M_r\) converges smoothly to a minimal hypersurface \(M_\infty\) in \(\mathbb{R}^{n+1}\).

In particular, if we denote \(A_r\) to be the second fundamental form of \(M_r\), then for any fixed \(x\in M=M_1\), we have

\[ A_r(\frac{x}{r})\to A_\infty(0)=0 \]

On the other hand, we have

\[ A_r(\frac{x}{r})=rA(x) \]

which implies \(A(x)=0\). Hence, \(M\) is flat, so \(u\) is an affine function. ◻