ESTIMATES ON THE GRAPHING RADIUS OF SUBMANIFOLDS AND THE INRADIUS OF DOMAINS

RALPH HOWARD

DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA COLUMBIA, S.C. 29208, USA

                           HOWARD@MATH.SC.EDU

Abstract. Let M be a submanifold of R^{n} be a complete immersed submanifold of R^{n} so that the principle curvatures in all normal directions are ≤ 1. Then any open geodesic ball of radius π in M is embedded in R^{n}. Moreover M graphs over any of its tangent planes on a closed ball (in the tangent plane) of radius 1. Several related estimates are given and used to prove inradius estimates for domains D ⊂ R^{n} whose boundaries have all principle curvatures in the interval [−1, 1]. For example the theorem of Lagunov any such domain with ∂D connected must have inradius at least 2/\sqrt{3} − 1 ≈ .15470038 … is proven.

Contents

1. Introduction 1 2. Curves of Bounded Curvature 2 3. Submanifolds of Bounded Curvature 4 4. Applications to Gradient Estimates and Reverse Isoperimetric

Inequalities 8 5. Applications to Inradius Estimates 10 References 15

1. Introduction

This notes are an attempt to extend some of the results of [5] to higher dimensions and in particular to prove the conjecture that, with the notation below, that if D ⊂ R^{n} is a bounded domain which is starlike with respect to the origin and so that all principle curvatures λ_{i} of ∂D satisfy |λ_{1}| ≤ 1, then the inradius of D is ≥ 1. Not much progress was made on this conjecture, and all the results here are in papers of Alexander and Bishop [2, 1], Howard and Treibergs [5], Lagunov [6, 7, 8], Lagunov-Fyet [9], and Pestov and Iomin [10]. Farther references can be found in these papers.

2 RALPH HOWARD

2. Curves of Bounded Curvature

Let c(t) be a unit speed curvature in R^{n} with curvature ≤ 1. That is |c′(t)| = 1 (2.1) and |c′′(t)| ≤ 1. (2.2) Choose coordinates so that c(0) = 0 and c′(0) = e_{1} (where e_{1}, …, e_{n} is the standard orthonormal basis of R^{n}). Write c(t) as c(t) = (x(t), y(t)) ∈ R × R^{n−1}. (2.3) Where x(t) ∈ R and y(t) ∈ R^{n−1} and

(2.4)  c(0) = (x(0), y(0)) = (0, 0)
(2.5)  c′(0) = (x′(0), y′(0)) = (1, 0)
(2.6)  |c′(t)| = x′(t)^{2} + |y′(t)|^{2} = 1
(2.7)  |c′′(t)| = x′′(t)^{2} + |y′′(t)|^{2} ≤ 1

Proposition 2.1. With notation as above we have for 0 ≤ t ≤ π

(2.8)  x′(t) ≥ cos(t)
(2.9)  |y′(t)| ≤ sin(t)
(2.10)  x(t) ≥ sin(t)
(2.11)  y(t) ≤ 1 − cos(t)

If equality holds at t = t_{0} in any one of these then for some unit vector u⊥e_{1} c(t) = cos(t)e_{1} + sin(t)u (2.12) holds for 0 ≤ t ≤ t_{0}. That is the segment of the curve between 0 and t_{0} is a segment of a great circle of a sphere tangent to e_{1} at the origin. Proof. Because x′(t)^{2}+|y′(t)|^{2} = 1 there is a continuous function θ(t) defined on the real line with θ(0) = 0 and x′(t) = cos(θ(t)), |y′(t)| = sin(θ(t)) (2.13) Note that because of the norm on y′ the function need not be smooth, however it is Lipschitz and therefore differentiable almost everywhere and the usual rules of calculus work almost everywhere. Define a plane curve γ by γ(t) = (cos(θ(t)), sin(θ(t))) = (x′(t), |y′(t)|) (2.14) Whence at all points where y′ ≠ 0

γ′(t) = (− sin(θ(t)), cos(θ(t)))θ′(t) = (x′′(t), \frac{⟨y′(t), y′′(t)⟩}{|y′(t)|}) (2.15)

and so at these points we can use (2.7)

|γ′(t)|^{2} = (θ′(t))^{2} = (x′′(t))^{2} + \frac{⟨y′(t), y′′(t)⟩^{2}}{|y′(t)|^{2}} ≤ (x′′(t))^{2} + |y′′(t)|^{2} ≤ 1 (2.16)

At points where both y′ and y′′ vanish it is not hard to see that θ′ = 0. This leaves the points where y′ = 0, but y′′ ≠ 0. At these points the derivative θ′ does not exist and as θ′ exists almost every where this is a set of measure zero. Therefore we have

|θ′(t)| ≤ 1 (2.17)

almost everywhere. This can be integrated to give

θ(t) ≤ t (2.18)

for all t ≥ 0. This implies

(2.19)  x′(t) = cos(θ(t)) ≥ cos(t)  for  0 ≤ π

and if equality holds at t = t_{0} then x′(t) = cos(t) on all of [0, t_{0}]. This prove the first part of the proposition.

For the second part note that

|y′|^{2} = 1 − (x′)^{2} ≥ 1 − cos^{2}(t) = sin^{2}(t). (2.20)

The last two parts now follow by integration.

Proposition 2.2. If c is a curve in R^{n} with curvature ≤ 1 and length ≤ 2π, then c is imbedded unless it has length 2π and is a standard circle. Moreover the endpoints of c satisfy the estimate (2.21) below.

Proof. Let L be the length of c, and choose a parameterization c : [−L/2, L/2] → R^{n} = R × R^{n−1} with c(0) = (0, 0) and c′(0) = (1, 0). With the notation as above we get c(t) = (x(t), y(t)) will satisfy

|c(L/2) − c(−L/2)| ≥ |x(L/2) − x(−L/2)| ≥ 2 sin(L/2) (2.21)

Proposition 2.3. If a curve c of length L ≤ π has curvature ≤ 1 and is tangent to a unit sphere at one of its end points, then c is disjoint from the interior of the sphere. If some point of the curve other that the end point is on the sphere, then the part of the curve between this point and the endpoint of tangency is a part of a great circle of the sphere.

Proof. We use the notation of the first proposition. Then c(0) = (0, 0) and c′(0) = (1, 0). Therefore the center of the sphere is at a point (0, u) for some unit vector u ∈ R^{n−1}. Thus

|c(t) − (0, u)|^{2} = x(t)^{2} + |u − y(t)|^{2}
≥ x(t)^{2} + (|u| − |y(t)|)^{2}
≥ sin^{2}(t) + (1 − 1 + cos(t))^{2}

= 1 (2.22)

which is exactly the statement of the proposition.

3. Submanifolds of Bounded Curvature

Let K = K(m, n) be the class m dimensional immersed submanifolds of R^{n} that have all principle curvatures in all normal directions ≤ 1. To be a little more precise let \overline{∇} be the usual flat connection on R^{n}, and for vector fields X, Y tangent to a submanifolds M^{m} of R^{n} split \overline{∇}_{X}Y into its tangent part ∇_{X}Y and the part II(X, Y) normal to T M. Then ∇ is the connection on M^{m} and II is the second fundamental form of M^{m} in R^{n}. The Gauss equation is then

\overline{∇}_{X}Y = ∇_{X}Y + II(X, Y). (3.1)

With this notation the submanifold M^{m} is in K if and only if

|II(X, Y)| ≤ |X||Y | (3.2)

for all vectors tangent to M^{m}. Note then for a curve c in M^{m} this leads to the relation

c′′ = \overline{∇}_{c′}c′ = ∇_{c′}c′ + II(c′, c′) (3.3)

A unit speed curve in M^{m} is a geodesic if and only if ∇_{c′}c′ = 0 so we have

Proposition 3.1. If c is a unit speed geodesic in M^{m} ∈ K then the curvature of c as a space curve is ≤ 1.

If X and Y are orthogonal unit vectors tangent to M^{m} at some point and K(X, Y) is the sectional curvature of the two plane spanned by X and Y then the Gauss curvature equation is

K(X, Y) = ⟨II(X, X), II(Y, Y)⟩ − |II(X, Y)|^{2} (3.4)

This leads to the following:

Proposition 3.2. The sectional curvatures of a manifold M in Ksatisfy

−2 ≤ K_{M} ≤ 1 (3.5)

Remark 3.3. These estimates are sharp. The upper is exact on the unit sphere. The Lower bound is sharp on complex curves in C^{2} = R^{4} at points where the Gauss curvature is −2. This second observation is due to Anton Schep.

Proof. Let P be a two plane tangent to M and let X, Y be an orthonormal basis of P. Then from the Gauss curvature equation

K(X, Y) = ⟨II(X, X), II(Y, Y)⟩ − |II(X, Y)|^{2}
≤ ⟨II(X, X), II(Y, Y)⟩
≤ |II(X, X)||II(Y, Y)|
≤ |X|^{2}|Y |^{2}

≤ 1 (3.6)

which proves the upper bound.

To prove the lower bound first note that

(3.7)  4|II(X, Y)| = |II(X + Y, X + Y) − II(X − Y, X − Y)|

≤ |X + Y |^{2} + |X − Y |^{2} = 2 + 2 = 4 (3.8)

and so |II(X, Y)| ≤ 1

K(X, Y) = ⟨II(X, X), II(Y, Y)⟩ − |II(X, Y)|^{2}
≥ −|II(X, X)||II(Y, Y)| − 1^{2}

≥ −1 − 1 = −2 (3.9)

Proposition 3.4. If M^{m} is in K and the geodesic ball of radius π about x_{0} ∈ M^{m} is complete, then the injectivity radius of M^{m} at x_{0} is ≥ π.

Proof. We argue that the cut locus C of x_{0} in M^{m} is disjoint from B(x_{0}, π). Any point x of C is either a focal point or there are two or more minimizing geodesics form x_{0} to x. If x is a focal point then by proposition 3.2 the sectional curvature is ≤ 1 and so by a standard comparison theorem the distance from x to x_{0} is at least π and thus x is not in B(x_{0}, π). Let x be the point of C closest to x_{0}, and assume (toward a contradiction) that x is in B(x_{0}, π). Then by a theorem of Klingenberg (cf. [3, lemma 5.6 p. 95] there is a a geodesic segment γ starting and ending at x_{0} that has x as a midpoint. However the length of this is less than 2π and by proposition 3.1 the curvature of γ as a space curve is ≤ 1. This contradicts proposition 2.2 that any such curve must be imbedded.

Proposition 3.5. (First Schur Lemma for Submanifolds) Let M^{m} ∈ K and let x_{0} ∈ M^{m}. Assume that the closed geodesic ball of radius π about x_{0} in M^{m} is complete. Then the open ball B(x_{0}, π) in M^{m} is an imbedded submanifold of R^{n}. If a unit sphere S^{n−1} of R^{n} is tangent to M^{m} at x_{0} then the ball B(x_{0}, π) is disjoint from the interior of this sphere. If some point x of B(x_{0}, π) is on S^{n−1} then the geodesic segment \overline{x_{0}x} of M^{m} also lies on S^{n−1} as a geodesic of S^{n−1}.

Remark 3.6. The proof of this given here (based on proposition 3.4) is due to Lars Andersson. I know of no other proof, however it is the type of thing that one expects to be well known to a certain school of Russian mathematicians.

Remark 3.7. In this result we mean that B(x_{0}, π) is imbedded in the strong sense that the map exp : T M_{x_{0}} → M^{m} ⊂ R^{n} is an injective immersion on the open ball of radius π in T M_{x_{0}}.

Proof. This follows from the the last proposition, propositions 2.1 and 2.3 and that geodesics of M^{m} have curvature ≤ 1.

Proposition 3.8. (Schur Lemma on the Graphing Radius) Let M^{m} ⊂ R^{n} be an immersed submanifold in the class K. Assume that M^{m} is tangent to R^{m} at the origin, and that the closed geodesic ball \overline{B}(0, π) of radius π about 0 is complete. Then there is a smooth function f : \overline{B}^{R^{m}}(0, 1) → R^{n−m} so that

Graph(f) := \{(x, f(x)) : x ∈ \overline{B}^{R^{m}}(0, 1)\} ⊆ \overline{B}^{M^{m}}(0, π) ⊆ M^{m} (3.10)

That is if M^{m} is in K then it is a graph over each of is tangent spaces on a disk of radius at least one.

Proof. The proof will only be give in the case of hypersurfaces. I know of no proof in the general case, but believe the ideas here, with some extra notational problems, should handle the higher codimension case also.

We can assume that M^{m} is tangent to R^{m} at the origin. (Where we have written R^{n} = R^{m+1} = R^{m} × {0}.) Then there is a neighborhood U of 0 so that M^{m} is a graph over U. That is there is a smooth function f : U → R with

f(0) = 0, df_{0} = 0 (3.11)

and so that

F (x) := (x, f(x)) (3.12)

is a parameterization of M^{m} near 0. We want to show that U can be taken to include the open unit ball of R^{m} centered at the origin.

Recall the basic imbedding invariants of M^{m} are give in this parameterization as follows: dF = (dx, df) and so the induced metric on M^{m} is given by

g = ⟨dF, dF ⟩ = |dx|^{2} + df^{2}

where |X| is the usual Euclidean length of a vector X ∈ R^{m}. The unit normal is

η(x) = \frac{(−∇f, 1)}{\sqrt{1 + |∇f|^{2}}} . (3.13)

The Hessian of F is given by

D^{2}F = (0, D^{2}f) (3.14)

and so the second fundamental form of the parameterization is

II = ⟨D^{2}f, η⟩ = \frac{D^{2}f}{\sqrt{1 + |∇f|^{2}}} (3.15)

As M^{m} is of class K we have the estimate |II(X, Y)| ≤ g(X, X)^{1/2}g(Y, Y)^{1/2}, whence,

\frac{|D^{2}f(X, Y)|}{\sqrt{1 + |∇f|^{2}}} ≤ g(X, X)^{\frac{1}{2}} g(Y, Y)^{\frac{1}{2}}
= (|X|^{2} + df(X)^{2})^{\frac{1}{2}} (|Y |^{2} + df(Y)^{2})^{\frac{1}{2}}
≤ ((1 + |∇f|^{2})|X|^{2})^{\frac{1}{2}} ((1 + |∇f|^{2})^{\frac{1}{2}} )

= (1 + |∇f|^{2})|X||Y | (3.16)

Let c(t) be a smooth curve in U and e_{1}, …, e_{m} an orthonormal basis for R^{m}. Then

\frac{d}{dt}|df_{c(t)}|^{2} = \frac{d}{dt} ∑ df_{c(t)}(e_{i})df_{c(t)}(e_{i})
= 2 ∑ df_{c(t)}(e_{i})D^{2}f(c′(t), e_{i})
= 2D_{c(t)}^{2} (c′(t), ∑ df_{c(t)}(e_{i})e_{i})

= D^{2}f(c′(t), ∇f) (3.17)

Which can be used to show that at points where df_{c(t)} ≠ 0 that

\frac{d}{dt}|df_{c(t)}| = D^{2}f (c′(t), \frac{∇f}{|∇f|}) (3.18)

In what follows we will use this formula along curves where it may be that df_{c(t)} = 0 at some points. When we do this in will always be the case that the result of the computation is going to be integrated. As the function t ↦ |df_{c(t)}| is Lipschitz it will always be differentiable almost everywhere, and the formula

∫_{a}^{b} \frac{d}{dt}|df_{c(t)}| dt = |df_{c(b)}| − |df_{c(a)}| (3.19)

will hold. This will be enough to get the estimates we need.

Now consider the function Φ : R → R given by

(3.20) Φ(s) = \frac{s}{\sqrt{1 + s^{2}}} so that Φ′(s) = \frac{1}{(1 + s^{2})^{\frac{3}{2}}}

Let u ∈ R^{m} be a unit vector and define a curve in R^{n} by c(t) = tu. Then c′(t) = u and so we can use (3.16) and (3.18)

|\frac{d}{dt}Φ(|df_{c(t)}|)| = |Φ′|df_{c(t)}|D^{2}f (u, \frac{∇f}{|∇f|})|

= \frac{|D^{2}f(u, |∇f|^{−1}∇f)|}{(1 + |∇f|^{2})^{\frac{3}{2}}}

≤ 1 (3.21)

where at the last step we have used that u and |∇f|^{−1}∇f| are both unit vectors.

But Φ(0) = 0 so (3.21) can be integrated to give

|Φ(df_{c(t)})| = \frac{|df_{c(t)}|}{\sqrt{1 + |df_{c(t)}|}} ≤ |t| = |c(t)| (3.22)

Setting x = c(t), this can be solved for |df_{x}| to give the inequality

|df_{x}| ≤ \frac{|x|}{\sqrt{1 − |x|^{2}}} . (3.23)

This estimate will hold on any ball B(0, r) with r < 1 provided that B(0, r) is contained in U. However note that if this holds on the closed ball of radius r < 1, then the tangent plane to M^{m} at the points over the boundary of B(0, r) do not have vertical tangent spaces as |df| is finite at these points. Thus the implicit function theorem lets us write M^{m} as a graph over a larger ball, and as long as the radius of this larger ball is less than one the estimate (3.23) will hold. Thus a straight forward analytic continuation argument shows that M^{m} is a graph over all of B(0, 1).

4. Applications to Gradient Estimates and Reverse Isoperimetric Inequalities

We now give some applications of the results of the last section.

Proposition 4.1. (Gradient Estimates for Hypersurfaces) Let D be a bounded domain in R^{n} and assume that the boundary M := ∂D is of class K. Define a function ρ : M → [0, ∞) by

ρ(x) = |x|^{2} (4.1)

and let ∇ρ be the gradient of ρ with respect to the induced metric on M. 1. If D contains the ball B(0, r) then

(4.2)  |∇ρ|^{2} ≤ (ρ − r^{2})((r + 2)^{2} − ρ)  whenever ρ ≤ r^{2} + 2r

2. If D is contained in B(0, R) with R > 2, then

(4.3)  |∇ρ|^{2} ≤ (R^{2} − ρ)(ρ − (R − 2)^{2})  whenever ρ ≥ R^{2} − 2R

Proof. To prove the first of these note that if (4.2) is false at a point x then both the unit spheres tangent to M = ∂D at x interest the interior of the ball B(0, r). (The picture is exactly the same as Figure 1 “Crash to earth” in the paper A reverse isoperimetric inequality, stability and extremal theorems for plane curves with bounded curvature). Now use the Schur lemma on the graphing radius to write M = ∂D as a graph over its tangent space at the point x. As both the unit spheres tangent to M at x interest B(0, r) and M is graph over the unit disk of the tangent space to M at these points that is sandwiched between these two spheres (by the first Schur lemma) this implies that M must also interest B(0, r), a contradiction.

The proof of the second inequality is similar. If (4.3) is false at the point x, then both the unit spheres tangent to M intersect the complement of B(0, R). Then the Schur lemma on the graphing radius will imply that M = ∂D also intersects the complement of B(0, R).

Remark 4.2. This result is true for higher codimension submanifolds of R^{n} in the following form: If M^{m} is a compact submanifold of R^{n} of class K and M^{m} is disjoint from the ball B(0, r) (resp. contained in B(0, R)) then the inequality (4.2) (resp. inequality (4.3)) holds (where ρ is defined as before). The proof is exactly as above.

Corollary 4.3. If D is a domain in R^{n} with boundary of class K and for some R < 3 we have

B(0, 1) ⊆ D ⊆ B(0, R), (4.4)

then D is starlike with respect to the origin.

Proof. Let M = ∂D. If D is not starlike with respect to the origin then there is a point x of M where the (affine) tangent space T M_{x} passes through the origin. At such a point the gradient of the restriction to M of the function x ↦ |x| has norm one. This implies that the function ρ(x) = |x|^{2} satisfies

|∇ρ|^{2} = 4ρ. (4.5)

But if x ∈ M then either |x| < \sqrt{3} (i.e. ρ(x) < 3) in which case (4.2) with r = 1 implies

|∇ρ|^{2} ≤ (ρ − 1)(9 − ρ) < 4ρ when ρ < 3 (4.6)

or \sqrt{3} ≤ |x| ≤ R (i.e. 3 ≤ ρ(x) ≤ R^{2}) then by (4.3) (and using both R < 3 and 3 ≤ ρ)

|∇ρ|^{2} ≤ (R^{2} − ρ)(ρ − (R − 2)^{2}) < 4ρ. (4.7)

This means that it is impossible for (4.5) to hold and therefore D must be starlike with respect to the origin.

Proposition 4.4. (Reverse Isoperimetric Inequality) Let D be a bounded domain in R^{n} whose boundary is of class K. Let A be the surface area of ∂D and V the volume of D. If D is starlike with respect to the origin then the inequality

A ≤ nV (4.8)

holds. There are also sharp lower bounds for A and V

A ≥ A(S^{n−1}) V ≥ V (B(0, 1)) (4.9)

Equality in any one of these inequalities implies D is a ball of radius one.

Proof. Let H be the mean curvature of ∂D with respect to the outward unit normal n, and let p(x) = ⟨x, n⟩ be the support function of ∂M. Then the Minkowski formulas

A = − ∫_{∂D} Hp dA, nV = ∫_{∂D} p dA (4.10) hold. As D is starlike the function p is positive, and as ∂D is in K the mean curvature satisfies |H| ≤ 1. Therefore

A = − ∫_{∂D} Hp dA ≤ ∫_{∂D} p dA = nV. (4.11)

If equality holds then H ≡ −1, which implies that ∂D is a sphere of radius one.

The isoperimetric inequality in R^{n} is

A(S^{n−1})^{n}V (D)^{n−1} ≤ V (B(0, 1))^{n−1}(A(∂D))^{n} (4.12)

Using the relation A(S^{n−1}) = nV (B(0, 1)) and the inequalities (4.8) and (4.12) leads at once to (4.9). If equality holds in (4.9) then it must also hold in (4.8) and so again D must be a ball of radius one.

5. Applications to Inradius Estimates

Proposition 5.1. Let D be a domain in R^{n} with boundary of class K. Assume that the interior of the ball B(0, r) is contained in D, and that ∂B(0, r) is tangent to ∂D at the points x_{1} and x_{2}. If r < 1 then the unit spheres tangent to D at x_{1} and x_{2} are disjoint.

Proof. Let S_{i}^{n−1} be the unit sphere tangent to D at x_{i}. Assume toward a contradiction that these spheres intersect. Then in the figure to the right let the small circle represent the ball B(0, r) and let S_{1}^{n−1} and S_{2}^{n−1} be as shown. Then by the the first Schur lemma (proposition 3.5) the boundary ∂D must stay between the sphere S_{i}^{n−1} and the sphere tangent to the boundary ∂D at x_{i}. Then using the lemma on the graphing radius (proposition 3.8) we see that the two “branches” of ∂D, the one tangent to S_{1}^{n−1} and the

S_{2}^{n−1} S_{1}^{n−1}

piece tangent to S_{2}^{n−1} must interest. Let c be the minimizing geodesic segment in ∂D from x_{1} to x_{2}. Then the length of c is less that π, but both ends of c are on the sphere ∂B(0, r). By the results of section 2 this is impossible. This contradiction completes the proof.

We record the following well known results:

Proposition 5.2. If D is a domain in R^{n} then the cut point of x ∈ ∂D is the point y along the inner normal to ∂D where x stops being the of ∂D closest to y. The locus C of D is the is the set all cuts points of D.

1. Every point x in the cut locus is either a focal point of ∂D or there are two or more minimizing segments form x to ∂D. If x is not a focal point then the number of minimizing geodesics to the boundary is finite.

2. For a bounded domain D the cut locus C is a strong deformation retract of D.

3. The map that takes a point y on the boundary to the cut point on the normal to ∂D at y is continuous.

Define R_{1} to be the radius of the circle at the center of an equilateral triangle with sides of length 2 and which is tangent to the three unit circles at the vertices of this triangle. Then R_{1} = 2/\sqrt{3} − 1 ≈ .15470038

Theorem 5.3 (Alexander-Bishop [1], Lagunov [6, 7, 8], Lagunov-Fyet [9]). Let D be a connected domain in R^{n} with boundary ∂D of class K. Assume that the inradius of D is less that R_{1} as defined above. Then the cut locus C of D is a smooth hypersurface of R^{n} and the natural map from ∂D to C is a double cover of C. There are the two cases

1. The boundary ∂D has two components. Then D is homeomorphic to the product C × [0, 1]

2. The boundary ∂D is connected. Then there is a double cover D̂ of D

that is diffeomorphic to C × [0, 1]. And in fact this latter case can never arise so that D is homeomorphic to a product C × [0, 1]. (The reason the result is stated with the extra case is that in the case of manifolds with boundary (which is what Alexander and Bishop consider) this case does come up.)

Proof. Let x ∈ C. Then x can not be a focal point of ∂D as ∂D is of class K and therefore any focal point is at a distance of at least one from ∂D, contradicting that the inradius of D is less that R_{1} < 1.

We now claim that there are exactly 2 minimizing segments from x to ∂D. There are at least two by proposition 5.2 If there were 3 or more such segments then let r = dist (x, ∂D) y_{1}, y_{2}, and y_{3} be three points of ∂D that can be connected to x by minimizing segments. Then ∂B(x, r) is tangent to ∂D at these three points. By proposition 5.1 the spheres S_{i}^{n−1} tangent to ∂B(x, r) at the y_{i}’s are disjoint. But this is impossible as r < R_{1}.

Now for x in C let y_{1} and y_{2} be the two point that are at a distance of r = dist (x, ∂D) from x. Let U_{i} be a very small open piece of ∂D containing y_{i}, and let ρ_{i}(z) = dist (z, U_{i}). Then the functions ρ_{1} and ρ_{2} are smooth in a neighborhood of x. Also the cut locus is local defined near x by the equations f := ρ_{2} − ρ_{1} = 0. The gradient of f is ∇f = ∇ρ_{2} − ∇ρ_{1} and at x this is just the difference of the two unit vectors u_{i} := (x − y_{i})/|x − y_{i}|. This is now zero unless y_{1} = y_{2}, which is not the case. Therefore ∇f(x) ≠ 0 and so by the implicit function this implies that the set C = {f = 0} is smooth near x. As x was any point of C we see that C is a smooth hypersurface.

This implicit function theorem argument also shows that the map that sends y ∈ ∂D to the point of C that is on the normal to ∂D at y is a local diffeomorphism (this uses that the inradius is less that the focal distance). As this map is exactly two to one it must be a covering map. This ∂D is a double cover of C.

Now at each point x ∈ C there are two segments connecting x with the boundary of D. Call the union of these two segments L_{x}. Then L_{x} is homeomorphic to the segment [−1, 1], with the point x on L_{x} corresponding to the point 0 of [−1, 1]. More over D is a disjoint union of the L_{x}’s and if x_{1} ≠ x_{2} then the segments L_{x_{1}} and L_{x_{2}} are disjoint. Therefore D is homeomorphic to a unit ball (i.e. a unit segment) bundle of a line bundle over C. This implies that the two cases given are the only ones that can come occur.

Finally we show that the case of ∂D connected can never occur. In all case we have that C is a connected smooth imbedded hypersurface of R^{n}. Therefore by a standard result of differential topology R^{n}\C into exactly two connected components (cf. [4, §4.4 pp. 103–108]) Moreover for each x ∈ C the segment L_{x} intersects C exactly once and this intersection is transverse so the endpoints of L_{x} are in different connected components of R^{n} \ C. However these endpoints are on ∂D and ∂D is disjoint from C so that if ∂D where connected we would have the contradiction that the endpoints of L_{x} were in the same connected component of R^{n} \ C. This completes the proof.

Corollary 5.4 (Lagunov [6, 7]). Let D ⊂ R^{n} be a connected domain with ∂D ∈ K and so that either ∂D is connected or ∂D has three or more connected components. Then the inradius of D is at least R_{1} = 2/\sqrt{3} − 1 ≈ .15470038 . . ..

Proof. This follows from the last theorem.

Let x be in the cut locus C of D. We say that x has order k if and only if x is not a focal point of ∂D and there are exactly k minimizing segments connecting x to ∂D. Note by the the first part of proposition 5.2 any non-focal point has a finite order ≥ 2.

Let R_{2} be the radius of the sphere at the center of an equilateral tetrahedron with sides of length two that is tangent to the four unit spheres centered on the vertices of the tetrahedron. Then R_{2} = \sqrt{3/2} − 1 ≈ .2247448714.

Proposition 5.5 (Lagunov [6, 7, 8], Lagunov-Fyet [9]). Let D be a domain in R^{n} with boundary of class K. If the inradius of D is less than R_{2}, then every point of the cut locus C of D is of order ≤ 3.

Proof. If x in the cut locus C of D has order four or more, then there are four points y_{i} ∈ ∂D that can be joined to x by minimizing segments. Let r = dist (x, ∂D). Then the ball B(x, r) is tangent to ∂D at the points y_{i}. By proposition 5.1 the spheres tangent to B(x, r) are disjoint. But this implies that r ≥ R_{2}. This contradiction completes the proof.

r 1 1

Let S(x_{i}, 1), i = 1, 2 be two unit spheres tangent to the ball B(0, r) of radius one centered at the origin. If the two spheres S(x_{i}, 1) are disjoint then the angle θ between x_{1} and x_{2} satisfies

sin \frac{θ}{2} ≥ \frac{1}{1 + r} (5.1)

as can easily be seen from the figure.

Theorem 5.6 (Lagunov [6, 7, 8], Lagunov-Fyet [9]). Let D be a domain in R^{n} with boundary of class K and inradius less that R_{2} defined above. Then the cut locus C of D is the disjoint union of the set C_{2} of cut points of order and the set C_{3} of points order three. The set C_{2} is a smooth imbedded hypersurface of R^{n}. The set C_{3} a smooth compact submanifold of R^{n} of dimension n − 2. Near any point x ∈ C_{3} the cut locus C has a neighborhood U in R^{n} so that C ∩ U is homeomorphic to a union of three closed half R^{n−1}’s glued together along a common R^{n−2}.

Before going on with the proof we give more detail about the normal form of the cut locus near the “singular” set of cut points of order three.

Let x_{1}, x_{2}, …, x_{n} be the standard coordinates on R^{n}. Let H_{1} = {x_{1} = 0, x_{2} ≥ 0}, H_{2} = {x_{2} = 0, x_{1} ≥ 0} and H_{3} = {x_{2} − x_{1} = 0, x_{1} ≤ 0}. Then near cut points of order three C locally looks like H_{1} ∪ H_{2} ∪ H_{3} near the origin. Thus C_{3} looks locally like the subspace {x_{1} = x_{2} = 0}.

Proof. By proposition 5.5 every cut point is of order at most three. That the set C_{2} is a smooth hypersurface in R^{n} follows exactly as in the proof of proposition 5.3. Now let x ∈ C_{3}. By a translation we can assume that x = 0 is the origin of R^{n}. Then let y_{1}, y_{2}, and y_{3} be the three points on ∂D that can be connected to x = 0 by a minimizing geodesic, and let r = dist (0, ∂D) = |y_{i}|. As in the proof of theorem 5.3 choose a very small open piece P_{i} of ∂D near y_{i} and let ρ_{i}(z) := dist (z, P_{i}). Then each function ρ_{i} is smooth in an neighborhood of 0 and ∇ρ_{i}(0) = −y_{i}/r. Let

f_{1} = ρ_{2} − ρ_{3},  f_{2} = ρ_{1} − ρ_{3},  f_{3} = ρ_{1} − ρ_{2}. (5.2)

Then the set C_{3} is defined near x by the equations f_{1} = f_{2} = f_{3} = 0 (as these equations are equivalent to ρ_{1} = ρ_{2} = ρ_{3}, that is the set of point that have three minimizing geodesics connecting them to the boundary.) But f_{3} = f_{2} − f_{1} so these equations can be replace by the pair of equations f_{1} = 0, f_{2} = 0. Therefore if we can show that the gradients of f_{1} and f_{2} are independent at x then the implicit function theorem will imply that the set C_{3} is a smooth codimension two submanifold of R^{n}. Let u_{i} := y_{i}/r. Then at x

∇f_{1} = u_{3} − u_{2}, ∇f_{2} = u_{3} − u_{2} (5.3)

If these two vectors are independent, then the vectors u_{1}, u_{2}, and u_{3} are dependent and thus all lie in some two dimensional plane. By the inequality 5.1 the angle θ between any two of these vectors satisfies

θ ≥ 2 sin^{−1} (\frac{1}{1 + R_{2}} ) > 109^{∘} (5.4)

But this implies that the figure formed by u_{1}, u_{2}, and u_{3} is rather rigid. The angle between the vectors u_{1} and u_{3} must lay between approximately 109^{∘} and 360^{∘} − 2 · 109^{∘} = 141^{∘}. As the vectors u_{i} are unit vectors this implies that the vectors u_{1}−u_{3} and u_{2}−u_{3} are linearly independent. This is equivalent to the independence of the vectors ∇f_{1} and ∇f_{2} at the point x = 0. This completes the proof that C_{3} is a smooth codimension two submanifold of R^{n}.

[figure: u_{3}; u_{1}; u_{2}; 109^{∘}; 141^{∘}]

Near the point 0 = x the part of C not in C_{3} is in one of the three sets

S_{1} = {ρ_{2} = ρ_{3} < ρ_{1}} = {f_{1} = 0, f_{2} < 0}
S_{2} = {ρ_{1} = ρ_{3} < ρ_{2}} = {f_{2} = 0, f_{1} < 0}
S_{3} = {ρ_{1} = ρ_{2} < ρ_{3}} = {f_{3} = 0, f_{1} > 0} = {f_{2} − f_{1} = 0, f_{1} > 0}.

But ∇f_{1} and ∇f_{2} are linearly independent at the point x = 0 and therefore they can be completed to a coordinate system f_{1}, f_{2}, h_{3}, … h_{n} centered at the point x = 0. This shows not only that locally near x that C is homeomorphic to the set described above, but that it is equivalent it the strong sense that for any point x ∈ C_{3} there is a local diffeomorphism ϕ of R^{n} that moves a neighborhood of x go a neighborhood of the origin and so that ϕ[C] coincides with the model set described above.

As an application of the last theorem we compute the Euler characteristic χ(C_{3}) of the singular set of the cut locus C of a domain D with boundary D of class K and with inradius as defined above. Let f : ∂D → C be the natural map, that is it sends a point y on the boundary to the point of C where the normal geodesic to ∂D intersects C. Choose a smooth triangulation of the submanifold C_{3} and extend it to a triangulation of C. Then (using that the map f is locally one-to-one) we can lift the triangulation of C to one of ∂D. (In doing this we may have to refine the triangulation of C so that each cell is inside a set where we can find a local inverse to f on this cell (this is not saying that we can find a local inverse to f near all points of C, but only that on cells of a properly chosen triangulation there will be an inverse)).

Let c_{i} denote the number of i-dimensional cells in the triangulation of ∂D. Then

χ(∂D) = ∑(−1)^{i}c_{i} (5.5)

For each cell σ of this triangulation either f[σ] ⊂ C_{3} or f[σ] is disjoint from C_{3}. (For this to be true I guess we are working with open cells.) Let c_{i}^{3} be the number of i-dimensional cells in ∂D that map into C_{3} and let c_{i}^{2} be the number of i-dimensional cells that have image under f disjoint from C_{3}. Note that by the definition of C_{3} each point x ∈ C_{3} has exactly three preimages under f and by thus the number of i-dimensional cells in C_{3} is (1/3)c_{i}^{3} and thus

χ(C_{3}) = \frac{1}{3} ∑(−1)^{i}c_{i}^{3}. (5.6)

Each point of C_{2} := C\C_{3} has exactly two preimages under f and so the total number of i-dimensional cells in C is c_{i}^{2}/2 + c_{i}^{3}/3 and thus the Euler characteristic of C is

χ(D) = χ(C) = \frac{1}{2} ∑(−1)^{i}c_{i}^{2} + \frac{1}{3} ∑(−1)^{i}c_{i}^{3} (5.7)

where χ(D) = χ(C) as C is a deformation retract of D. Finally there is the obvious relation

c_{i} = c_{i}^{2} + c_{i}^{3} (5.8)

The equations (5.5), (5.6), (5.7), and (5.8) imply:

Proposition 5.7. With the notation above

2χ(D) = χ(∂D) − χ(C_{3}) (5.9)

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