The discrete Gauss-Bonnet theorem

This is a slight extension of my previous note on discrete Gauss-Bonnet theorem. As mentioned in that note, this is a generalization of the well-known fact that the sum of the exterior angles of a polygon is always ${{2\pi}}$, which can also be regarded as a very special case of the Gauss-Bonnet theorem.

First of all we introduce a discrete version of surfaces. Let ${ M}$ be a (${2}$-dimensional) “discrete surface” (here for examples) or a simplicial surface (with or without boundary), which for simplicity is assumed to be contained in ${\mathbb R^n}$. A discrete surface consists of finitely many “triangles” gluing along their edges. Each triangle is isometric to an ordinary planar triangle on ${\mathbb R^2}$. We call each triangle a “face”. For each face, there are exactly three edges. And for each edge, there are exactly two vertices. If two different faces intersect, we assume they either intersect at their common edge or their common vertex. Clearly the discrete notion of surfaces is very useful in computer graphics.

A discrete surface

A more complicated discrete surface

The main result of this post is

 Theorem 1 (Discrete Gauss-Bonnet theorem) Let ${K}$ be the Gaussian curvature and ${k_g}$ be the geodesic curvature on a discrete surface ${M}$, then $\displaystyle \displaystyle \sum_{v\in \boldsymbol{V_I}} K(v)+\sum_{v\in \boldsymbol{V_B}}k_g(v)=2\pi\chi(M).$ Here ${\boldsymbol{V_I}}$ is the set of interior vertices and ${\boldsymbol{V_B}}$ is the set of boundary vertices and ${\chi(M)}$ is the Euler characteristic of ${M}$.

We will explain the notation below the fold.

Our first goal is to generalize all the notions such as boundary, geodesic curvature (or exterior angle at an vertex of a triangle), and Gaussian curvature on a smooth surface to a discrete surface. Continue reading

Why a vector field rotates about its curl?

Let ${F= F(x_1, x_2, x_3): \mathbb R^3\rightarrow \mathbb R^3}$ be a smooth vector field on ${\mathbb R^3}$. We define its curl to be the vector field

$\displaystyle \textrm{curl } F:= \left(\frac{\partial F_3}{\partial x_2}- \frac{\partial F_2}{\partial x_3}, \frac{\partial F_1}{\partial x_3}- \frac{\partial F_3}{\partial x_1}, \frac{\partial F_2}{\partial x_1}- \frac{\partial F_1}{\partial x_2} \right) \ \ \ \ \ (1)$

It is often claimed in advanced calculus that ${\textrm{curl }F(p)}$ measures how fast ${F}$ “rotates”. More specifically, suppose ${\textrm{curl }F(p)\ne 0}$, then locally around ${p}$, the vector field ${F}$ behaves like a “swirl”, rotating (in the anti-clockwise sense) around the axis pointing in the same direction as ${\textrm{curl }F(p)}$ and with angular speed ${|\textrm{curl }F(p)|}$. (Strictly speaking this picture is not quite true even locally: ${\textrm{curl } F(p)}$ only captures the “rotational” part of ${F(x)}$ around ${p}$. Besides the constant part ${F(p)}$ and the “rotational” part ${\textrm{curl }F(p)}$, its first order part also contain the “flux” part which is given by ${\textrm{div } F(p)}$, but let’s ignore this for the discussion here.)

To my dismay, while this gives a good geometric description of ${\textrm{curl }F}$, I find that in many textbooks the demonstration of this “fact” is either by looking at the curl of some very special vector field (like ${(-y, x, 0)}$), or using the Stokes’ Theorem. The first approach is really too simplistic although it gives a good model for a “rotational” vector field. The second approach, while rigorous, is “a posteriori”, as it seems that the rather obscure definition of curl given in (1) just comes from nowhere. How on earth would one come up with such a definition at the very beginning? Continue reading

Posted in Calculus, Linear Algebra | 1 Comment

A functional inequality on the boundary of static manifolds

1. introduction and statement of results

The research in this article is largely motivated by the following result concerning a functional inequality on the boundary of bounded domains in the Euclidean space ${\mathbb R^n}$, proved in [MTX] Corollary 3.1.

 Theorem 1 ([MTX]) Let ${ \Omega \subset \mathbb R^n }$ be a bounded domain with smooth boundary ${\Sigma}$. Let ${H}$ and ${\mathbb{II}}$ be the mean curvature and the second fundamental form of ${\Sigma }$ with respect to the outward normal respectively. If ${H>0}$, then $\displaystyle \int_\Sigma \left[ \frac{ ( \Delta_{_\Sigma} \eta )^2 }{ H } - \mathbb{II} ( \nabla_{_\Sigma} \eta ,\nabla_{_\Sigma} \eta )\right] d \sigma \ge 0 \ \ \ \ \ (1)$   for any smooth function ${ \eta }$ on ${\Sigma}$. Here ${ \nabla_{_\Sigma} }$, ${ \Delta_{_\Sigma} }$ denote the gradient, the Laplacian on ${ \Sigma}$ respectively, and ${ d \sigma }$ is the volume form on ${ \Sigma }$. Moreover, equality in (1) holds for some ${ \eta }$ if and only if ${ \eta = a_0 + \sum_{i=1}^n a_i x_i }$ for some constants ${a_0, a_1, \ldots, a_n }$. Here ${\{ x_1, \ldots, x_n\}}$ are the standard coordinate functions on ${\mathbb R^n}$.

When ${n=3}$ and ${\Sigma}$ is convex, it is known ([MTX]) that the functional on the left side of (1) represents the second variation along ${\eta}$ of the Wang-Yau quasi-local energy ( [WY1], [WY2] ) at the ${2}$-surface ${ \Sigma }$, lying in the time-symmetric slice ${\mathbb R^3 = \{ t = 0 \}}$, in the Minkowski spacetime ${ \mathbb R^{3,1}}$. Thus, (1) can be relativistically interpreted as the stability inequality of the Wang-Yau energy at ${ \Sigma}$. The general case of such a stability inequality is implied by results in [CWY], [WY2] for a closed, embedded, spacelike ${2}$-surface in ${ \mathbb R^{3,1}}$ that projects to a convex ${2}$-surface along some timelike direction.

In this article, adopting a Riemannian geometry point of view, we generalize Theorem 1 to hypersurfaces that are boundaries of bounded domains in a simply connected space form. More generally, we give an analogue of (1) on the boundary of compact Riemannian manifolds whose metrics are static (see Definition 3).

First, we fix some notations. Given a constant ${\kappa > 0}$, let ${\mathbb H^n (\kappa)}$ and ${\mathbb{S}^n_+ ( \kappa )}$ denote an ${n}$-dimensional hyperbolic space of constant sectional curvature ${-\kappa}$ and an ${n}$-dimensional open hemisphere of constant sectional curvature ${\kappa }$ respectively.

 Theorem 2 Suppose ${(M , g)}$ is one of ${\mathbb{R}^n}$, ${\mathbb H^n (\kappa)}$ and ${\mathbb{S}^n_+(\kappa)}$. Let ${V }$ be the positive function on ${M}$ given by $\displaystyle V= \begin{cases} 1, \quad & \displaystyle \mathrm{if} \ (M, g) =\mathbb{R}^n,\\ \cosh \sqrt\kappa r, \quad & \displaystyle \mathrm{if} \ (M, g) =\mathbb{H}^n(\kappa), \\ \cos \sqrt\kappa r, \quad & \displaystyle \mathrm{if} \ (M, g) =\mathbb{S}^n_+(\kappa), \end{cases} \ \ \ \ \ (2)$   where ${r}$ is the distance function from a fixed point ${p}$ on ${(M, g)}$. When ${ (M, g) = \mathbb{S}^n_+ ( \kappa )}$, ${p}$ is chosen to be the center of ${\mathbb{S}^n_+ ( \kappa ) }$ so that ${V>0}$ on ${M}$. Given a bounded domain ${\Omega \subset M }$ with smooth boundary ${\Sigma}$, let ${H}$ and ${\mathbb{II}}$ be the mean curvature and the second fundamental form of ${\Sigma}$ respectively. If ${H > 0 }$, then for any smooth function ${\eta}$ on ${\Sigma}$, $\displaystyle \begin{array}{rl} & \displaystyle \int_\Sigma V \left[ \frac{ \left[ \Delta_{\Sigma} \eta + (n-1) k \eta \right]^2 }{H} - \mathbb{II} (\nabla_{\Sigma} \eta, \nabla_{\Sigma} \eta) \right] d \sigma \\ \ge & \displaystyle \int_\Sigma \frac{\partial V}{\partial \nu} \left[ | \nabla_{\Sigma} \eta |^2 - (n-1) k \eta^2 \right] d \sigma . \end{array} \ \ \ \ \ (3)$   Here ${k = 0 }$ or ${\pm \kappa}$ is the sectional curvature of ${(M,g)}$. Moreover, equality in (3) holds if and only if ${\eta}$ is the restriction of a function $\displaystyle u = \begin{cases} a_0 + \sum_{i=1}^n a_i x_i, & \displaystyle \mathrm{if} \ (M, g) =\mathbb{R}^n,\\ a_0 t + \sum_{i=1}^n a_i x_i, & \displaystyle \mathrm{if} \ (M, g) =\mathbb{H}^n(\kappa), \\ a_0 x_0 + \sum_{i=1}^n a_i x_i, & \displaystyle \mathrm{if} \ (M, g) =\mathbb{S}^n_+(\kappa) . \end{cases} \ \ \ \ \ (4)$   Here ${a_0, \ldots, a_n }$ are arbitrary constants, ${\mathbb H^n (\kappa)}$ is identified with $\displaystyle \left\{ (t, x_1, \ldots, x_n) \in \mathbb R^{n,1} \ | \ - t^2 + \sum_{i=1}^n x_i^2 = - \frac{1}{\kappa}, \ t > 0 \right\}$ in the ${(n+1)}$-dimensional Minkowski space ${\mathbb R^{n,1}}$ and ${\mathbb{S}^n_+ ( \kappa )}$ is identified with $\displaystyle \left\{ (x_0, x_1, \ldots, x_n) \in \mathbb R^{n+1} \ | \sum_{i=0}^n x_i^2 = \frac{1}{\kappa} , \ x_0 > 0 \right\}$ in the ${(n+1)}$-dimensional Euclidean space ${\mathbb R^{n+1}}$.

An extension of the First Fundamental Theorem of Calculus

In this note, we record a simple extension of the first fundamental theorem of calculus. Let us recall the first and second fundamental theorems of calculus:

 Theorem 1 (First fundamental theorem of calculus) If ${f:[a, b]\rightarrow \mathbb R}$ is continuous, then for ${x\in (a, b),}$ $\displaystyle \frac{d}{dx}\int_a^x f(t)dt= f(x).$
 Theorem 2 (Second fundamental theorem of calculus) If ${F:[a, b]\rightarrow \mathbb R}$ is ${C^1}$, then $\displaystyle \int_a^b F'(t)dt=F(b)-F(a).$

It is well-known that Theorem 1 implies Theorem 2. Indeed, assuming Theorem 1, and for ${F}$ in the assumption of Theorem 2, by letting ${G(x)= \int_a^x F'(t)dt}$ and using Theorem 1 to differentiate the difference ${G(x)-F(x)}$, we can easily get Theorem 2.

It is also well-known that Theorem 2 can be generalized to many situations (usually called Stokes theorem or divergence theorem). E.g. for a smooth ${(n-1)}$ form ${\omega}$ on a compact manifold ${M}$ with boundary ${\partial M}$, we have

$\displaystyle \int_M d\omega = \int_{\partial M}\iota^*\omega.$

Here ${\iota}$ is the inclusion of ${\partial M}$ in ${M}$.

It is therefore natural to ask if an extension of Theorem 1 exists in higher dimension. I will illustrate one such extension.

For simplicity let’s assume ${f:\overline \Omega\rightarrow \mathbb R}$ is continuous and ${h: \overline \Omega\rightarrow \mathbb R}$ is ${C^1}$, where ${\Omega}$ is a bounded open set in ${\mathbb R^n}$ and ${\overline \Omega}$ is its closure. Then we have

 Theorem 3 If ${H_{n-1}(h^{-1}(t))<\infty}$, then $\displaystyle \frac{d}{dt}\int_{\{x\in \Omega: h(x)\le t\}} f(x)|\nabla h(x)|dx= \int_{h^{-1}(t)}f(x)dH_{n-1}(x),$ where ${H_{n-1}}$ is the ${(n-1)}$-dimensional Hausdorff measure.

When ${n=1}$, ${\Omega=(a, b)}$ and ${h(t)=t}$, we can recover Theorem 1.
Proof: We have the coarea formula (which generalizes the Fubini’s theorem)

$\displaystyle \int_\Omega f(x)|\nabla h(x)|dx= \int_{-\infty}^\infty \int_{h^{-1}(t)} f(x)dH_{n-1}(x)dt.$

Apply this formula to ${\Omega_s =\{x\in \Omega: h(x)\le s\}}$, then we have

$\displaystyle \int_{\{x\in \Omega: h(x)\le s\}}f(x)|\nabla h(x)|dx= \int_{-\infty}^s \int_{h^{-1}(t)}f(x)dH_{n-1}(x)dt.$

Differentiating both sides with respect to ${s}$, and applying the first fundamental theorem of calculus (Theorem 1), we can get the result. $\Box$

Problem: Find the most general settings for this result to hold. Do we have other non-trivial generalization?

A discussion forum

In order to facilitate discussion, I have set up a discussion forum here:
http://cuhkmath.606h.net/phpBB3/viewforum.php?f=2

One of the most effective way of learning mathematics is by discussion. While there are excellent sites like MathOverflow or StackExchange where you may even find Fields medalists answering your questions, the purpose of this forum is that we could keep the discussion more “local” (if you are afraid of asking in front of the whole world), informal, and perhaps more down-to-earth, without feeling any pressure of asking “stupid questions”. Another advantage is the latex support here, which is better than other platforms, e.g. facebook. I hope we can gather a group of math students/mathematicians so that the forum can run effectively.

To participate in a discussion, please register for a login name (it’s free). Registered users can post new topics and reply on the forum. You will be assigned as an administrator once your identity is confirmed. Although I name it “cuhkmath”, it is open to everyone, so feel free to join!

Since I am not an expert in technology, feel free to suggest how we can improve this forum. (In fact I am still learning how to configure that site. ) E.g. I am using a free web hosting service (BinHoster) and it seems that the sever is not very stable sometimes. You may suggest how to migrate it to another site if you have a better option.

LaTex can be used in the forum, which is an advantage. I encourage you to make use of the preview function for testing.

Lastly, I want to say that it’s impossible to keep the forum alive by a single person. Your participation by actively participating in discussions is really really appreciated.

Thank you and enjoy! :D

Mean value properties for harmonic functions on Riemannian manifolds

It is well-known that a continuous function ${u}$ on ${\mathbb R^n}$ satisfies the mean value property over spheres if and only if it is harmonic i.e. ${\Delta u=0}$. More generally, this also holds on a Riemannian space form. However, this property is generally not true on a general Riemannian manifold. In fact, it is not hard to show that the mean-value property is true only when for each point ${p}$ of ${M}$, each geodesic sphere near ${p}$ has constant mean curvature ([Fr], [W]). Such a manifold is called a harmonic manifold.

This motivates us to investigate into how well the mean-value type property holds on a general Riemannian manifold, and how it may improve on an Einstein manifold (which is not necessarily harmonic).

In this note, we prove that (Theorem 1) for any smooth function ${u}$ on an ${n}$-dimensional Riemannian manifold ${(M,g)}$, it holds that near ${p}$, its average value over a geodesic sphere ${S_r(p)}$ satisfies

$\displaystyle \frac{1}{|S_r(p)|}\int_{S_r(p)}u dS_r= u(p)+ \frac{\Delta u(p)}{2n}r^2 +O(r^4). \ \ \ \ \ (1)$

Here ${|S_r(p)|}$ denotes the ${(n-1)}$-dimensional area of ${S_r(p)}$. Actually, the ${O(r^4)}$ term is explicit and has been given in Theorem 1. In particular, if ${u}$ is harmonic,

$\displaystyle \frac{1}{|S_r(p)|}\int_{S_r(p)}u dS_r= u(p)+O(r^4).$

When ${(M,g)}$ is Einstein, this can be improved to

$\displaystyle \frac{1}{|S_r(p)|}\int_{S_r(p)}u dS_r= u(p)+O(r^6). \ \ \ \ \ (2)$

These kinds of results have been previously obtained by Friedman [Fr], and also Gray-Willmore [GW]. Friedman showed that for any nonzero (and so locally either positive or negative) harmonic function ${u}$ on an Einstein manifold, it holds that near ${p}$,

$\displaystyle \frac{1}{|S_r(p)|}\int _{S_r(p)}u dS_r = u(p)+O(r^3).$

His approach involves directly differentiating ${\overline u(r)=\frac{1}{|S_r(p)|}\int _{S_r(p)}u dS_r }$ with respect to ${r}$, which would yield an integral involving the mean curvature ${H(S_r)}$ of ${S_r}$. He then examined the expansion of ${H(S_r)}$ to obtain the derivative estimate ${\frac{\overline u'(r)}{\overline u(r)}=O(r^2)}$, thus integrating with respect to ${r}$ would lead to his result. His method does not seem to be adaptable to the case where ${u(p)=0}$. Moreover, the order ${O(r^3)}$ seems to be not optimal.

On the other hand, Gray and Willmore [GW] Theorem 4.5 improved Friedman’s result when ${(M,g)}$ is an analytic Riemannian manifold: they showed that in this case, (1) and (2) are true. However, their approach is quite complicated and not very illuminating. Indeed, they proved the result by comparing the Riemannian Laplacian with the so called Euclidean Laplacian, which is just ${\sum_{i=1}^n \frac{\partial ^2}{\partial {x^i}^2}}$ in the normal coordinates. They were able to express their difference in terms of the curvature of ${M}$ and using various commutative formulas for the covariant derivatives, they can transfer the Euclidean mean-value property to the Riemannian setting, with some additional curvature terms. Unfortunately, the Euclidean Laplacian ${\widetilde \Delta}$ is not intrinsic to the geometry of ${M}$ (e.g. it is not true that when ${k\ge 2}$, ${(\widetilde \Delta)^k = \Delta^k}$, even at the point ${p}$), and the details are indeed quite hard to be checked completely.

In this note, we provide a somewhat more transparent proof of these kinds of results. Our main idea is that when ${r}$ is small enough, ${S_r}$ is sufficiently close to a Euclidean sphere, to the extent which enables us to use the symmetry of some spherical integrals to obtain a lot of cancelations and improve the error estimate of Friedman. Continue reading

Posted in Analysis, Calculus, Geometry | 2 Comments

“Useless” Circle Properties: The Green Flash In Mathematics

Since F.4 I’ve been thinking long and hard about real-life applications of circle properties. Why do we study them in secondary school anyway? Someone told me they’re used in designing cylindrical structures, but I couldn’t find a satisfactory book or website that explains how we apply circle properties just like we use modular arithmetic in time and date calculations. Therefore, upon discovering the Maths in the Workplace video, I got super excited and researched.

The following summarises my findings. I am no match for the masters on this blog (as of posting this article) so I’m posting a rather elementary article here. Mathematics has to be useful, I insist, and that why I’ve taken the road less travelled.