1 Some background in differential geometry

The goal of this chapter is developing some basics of differential geometry needed to investigate minimal surfaces in R3 and Rn.

1.1 Curves in ℝn

We begin with some basic notions. A parameterized curve in Rn is a map α:I→Rn, where I is an interval in R. It is said to be regular if α′(t)≠0 for all t∈I. We can write α(t)=(x1(t),x2(t),…,xn(t)). For example, α(t)=(cost,sint,t) for t∈R is a parameterization of a helix.

The tangent vector (or velocity vector) of α is α′(t)=(x1′(t),…,xn′(t)). The arc length of a curve on [a,b] is

A curve α(t) is of unit speed if ‖α′(t)‖≡1. Any nonunit speed regular curve α(t) can be reparameterized by changing the parameter t to the arc-length parameter s, i. e., s(t)=∫at‖α′(u)‖du, to form a unit speed curve β(s):=α(t(s)), where t=t(s) is the inverse of the function s=s(t). Therefore we can always assume that the curves we will be discussing are unit-speed curves. This assumption means that we are only interested in the geometric shape of a regular curve since reparameterizing does not change the shape of a curve.

Let α:I→R3 be a unit-speed curve. We are interested in the rate of change of the tangent vector. Denote by T(s):=α′(s) the unit-tangent vector. Then the curvature of α is

Denote N(s):=T′(s)‖T′(s)‖, the principal normal of α, and B(s)=T(s)×N(s), the binormal of α. We have the following Frenet formulas:

where τ(s) given by B′(s)=−τ(s)N(s) is the torsion of α.

The location of p does not matter because of the symmetry of the circle. A circle of radius r can be parameterized by α(t)=(rcost,rsint,0), 0<t<2π. Notice that this is not a unit-speed curve since α′(t)=(−rsint,rcost,0), and hence ‖α′(t)‖=r. So we reparameterize it into a unit-speed curve by letting sr=t to get β(s)=(rcos(s/r),rsin(s/r),0), 0<s<2πr. Therefore

Thus

that is, the curvature of a circle of radius r is 1r. Notice that if r is large, then the curvature is small, whereas if r is small, then the curvature is large.

1.2 Surfaces in ℝ3 and ℝn

A (local) parameterized surface in Rn is a one-to-one map x:U⊂R2→Rn, where U is an open subset of R2. In this book, we always assume that x is of class at least C2, that is, the partial derivatives of x up to the second order all exist and are continuous. We can write x(u,v)=(x1(u,v),x2(u,v),…,xn(u,v)),(u,v)∈U. The surface is said to be regular if xu×xv≠0 for all points (u,v)∈U. A global surface in Rn is a set M such that for every point p∈M, there is a local parameterization x:U⊂R2→M⊂Rn with p∈x(U).

We now focus on the surfaces M in R3 and introduce various notions of curvatures for M. A curve on a surface M is a mapping from an interval to the surface, that is, α:I⊂R→M⊂R3. We can write α(t)=x(u(t),v(t)) for some functions u,v:I→R. Given p∈M, a tangent vector to M at p∈M is the tangent vector of some curve α:(−ϵ,ϵ)→M (for some ϵ>0) with α(0)=p, that is, it is α′(0). The tangent space of M at p∈M, denoted by TpM, consists of all of the tangent vectors to M at p∈M, that is,

which is a vector space with standard basis {xu|p,xv|p}. The unit normal to the surface n is given by

The first fundamental form (also called a metric) I assigns to every p∈M the map Ip:TpM×TpM→R given by Ip(v,w)=v·w for all v,w∈TpM, where v·w is the standard dot product of v,w (as vectors in R3). If we write v=axu+bxv and w=cxu+dxv, then I(v,w)=v·w=(ac)g11+(ad+bc)g12+(bd)g22, where g11=xu·xu,g12=g21=xu·xv,g22=xv·xv. For this reason, we call g11,g12,g21,g22 the coefficients of the first fundamental form.

We now ready to introduce various notions of the curvatures of surfaces. As we have seen above, the definition of the curvature for curves used the (unit) tangent vector of the curves. However, in the surface case, dimTpM=2, so the choices of the tangent vectors for M are not canonical. Instead, we use its unit-normal n to define the curvatures. We use two ways to introduce various notions of curvatures for surfaces: (1) through curves on the surfaces and (2) through the shape operator Sp:=−dn|p. We first discuss the approach of using curves on the given surfaces.

The normal curvature of curves in a surface

Let α be a unit-speed curve on a surface M. We introduce the Darboux frame along the curve, {α′(t),n(t),n(t)×α′(t)}, where n(t) is the unit normal to the surface restricted to the α(t) (i. e., n(t)=n(α(t))). It is a moving orthonormal frame, so every other vector can be written as a linear combination of the vectors in the Darboux frame. Notice that α″ is perpendicular to α′, so we can write

where κn(t)=α″(t)·n(t),κg(t)=α″(t)·(n(t)×α′(t)). The scalars κn(t) and κg(t) are called the normal curvature and the geodesic curvature of α at the point p=α(t). We have κ2(t)=κn2(t)+κg2(t), where κ(t) is the curvature of α.

The normal curvature of surfaces with respect to a given tangent vector v∈TpM

Let M be a surface in R3, p∈M, and let v∈TpM be a unit tangent vector (direction). Then there is a (unit-speed) curve α:(0,ϵ)→M such that α(t0)=p and α′(t0)=v for some t0∈(0,ϵ). We define the normal curvature of M at p in the direction v, denoted by κn,v(p), as the normal curvature of the...