# Download e-book for kindle: Algebraic Curves: An Introduction to Algebraic Geometry by William Fulton By William Fulton

ISBN-10: 0201510103

ISBN-13: 9780201510102

ISBN-10: 0805330828

ISBN-13: 9780805330823

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11) |∇v(x)|2 |A(∇v(x))| ≤ const Ξ + |∇v(x)|2 a(|∇v(x)|) for any x ∈ RN . 12) BR \B√R Ξ + |∇v(x)|2 a(|∇v|) dx ≤ C ln R , |Y |2 as long as R is large enough. Then, given R > 0 (to be taken appropriately large in what follows) and x ∈ RN , we now define √  1 if |Y | ≤ R,  √ |) ϕR (x) := 2 ln(R/|Y if R < |Y | < R, ln R  0 if |Y | ≥ R. By construction, ϕR is a Lipschitz function and const x + v(x)∇v(x) ∇ϕR (x) = − |Y |2 ln R √ for any x ∈ RN such that R < |Y | < R. 13) ≤ const |Y |4 ln2 R A(∇v(x))∇ϕR (x) · ∇ϕR (x) A(∇v(x)) x · x + v 2 (x) A(∇v(x))∇v(x) · ∇v(x) .

Doris Fischer-Colbrie and Richard Schoen. The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. Comm. Pure Appl. , 33(2):199–211, 1980. N. Ghoussoub and C. Gui. On a conjecture of De Giorgi and some related problems. Math. , 311(3):481–491, 1998. David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. Elliott H. Lieb and Michael Loss.

Hence, there exists an interval (β2 , β3 ), which we suppose as large as possible, with β2 > β =: β1 and β3 ∈ (β2 , +∞) ∪ {+∞} in such a way that h′ (t) > 0 for any t ∈ (β2 , β3 ). 10, that we have already shown to be impossible. Therefore, β3 = +∞. Analogously, h(t) must be equal to m in [β1 , β2 ] because, if not there would be an interval (β1′ , β2′ ) ⊂ [β1 , β2 ] in such a way that h′ (t) = 0 in (β1′ , β2′ ) and h′ (β1′ ) = h′ (β2′ ) = 0, reducing again to the impossible case III. This shows that h′ (t) = 0 for t < β1 and t > β2 and h(t) = m for t ∈ [β1 , β2 ], yielding case E.