By J. W. Bruce

The differential geometry of curves and surfaces in Euclidean area has involved mathematicians because the time of Newton. the following the authors take a unique method by way of casting the speculation right into a new mild, that of singularity idea. the second one version of this winning textbook has been completely revised all through and incorporates a multitude of recent workouts and examples. a brand new ultimate bankruptcy has been additional that covers lately built suggestions within the type of features of a number of variables, a topic crucial to many purposes of singularity conception. additionally during this moment variation are new sections at the Morse lemma and the category of airplane curve singularities. the one must haves for college students to stick to this textbook are a familiarity with linear algebra and complex calculus. therefore will probably be worthwhile for somebody who would favor an creation to the trendy theories of catastrophes and singularities.

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**Extra resources for Curves and singularities: a geometrical introduction to singularity theory**

**Sample text**

Suppose now that Σk is an area-minimizing minimal hypersurface. 20) for minimal graphs in Chapter 1, the density VΣ is uniformly bounded. If rj → 0, then the sequence Σj = rj Σ of dilated surfaces is also areaminimizing. The bound on VΣj implies the existence of a convergent subsequence (say, as area-minimizing currents or stationary varifolds); see the next chapter for more on this. The (not necessarily unique) limit is denoted by Σ∞ and is also areaminimizing. It follows from the monotonicity formula that VΣ∞ = VΣ is also equal to the density of Σ∞ at the origin.

93) Cm = {(x1 , . . , x2m ) | x21 + · · · + x2m = x2m+1 + · · · + x22m } ⊂ R2m are area-minimizing. 3 Weak Bernstein-Type Theorems In this chapter, we will prove a generalization of the classical Bernstein theorem for minimal surfaces discussed in the previous chapters. The various Bernstein theorems imply that, through dimension seven, area-minimizing hypersurfaces must be affine. A weaker form of this is true in all dimensions by the Allard regularity theorem [Al1]. 1) Vol(Br (x) ∩ Σ) ≤ (1 + δ) Vol(Br ⊂ Rk ) for all x and r, then Σ is an affine k-plane.

Letting L(t) denote the length of ∂BtΣ , the usual comparison theorem for nonpositive curvature implies that L(t) ≥ 2 π t . 70) kg , ∂BtΣ where kg is the geodesic curvature. 68). 72) L(t) ≥ 2+ C 2 πt. 40)), Area(BtΣ ) = 0 L(t). 73) Area(B Σ 1 ) ≥ π 8 1+ C 32 1 . 73) for sufficiently small. This contradiction completes the proof. 3 implies the smooth version of Allard’s estimate. 16. This, and related results, are discussed in more detail in [CM8]. 3 Lp Bounds of |A|2 for Stable Hypersurfaces In this section, we combine Simons’ inequality with the stability inequality to show higher Lp bounds for the square of the norm of the second fundamental form for stable minimal hypersurfaces.