Curved Background Login Form Example - Utilises overdraw of the CustomPaint to draw the curve behind the content even when the keyboard appears.

main.dart

import 'package:flutter/cupertino.dart';
import 'package:flutter/material.dart';
import 'package:flutter/services.dart';

void main() => runApp(App());

class App extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      theme: ThemeData(
        brightness: Brightness.light,
        primaryColor: Color(0xFF3A455C),
        inputDecorationTheme: InputDecorationTheme(
          contentPadding: EdgeInsets.all(8.0),
          border: OutlineInputBorder(
            borderRadius: BorderRadius.circular(48.0),
          ),
        ),
        buttonColor: Color(0xFF3A455C),
        buttonTheme: ButtonThemeData(
          buttonColor: Color(0xFF3A455C),
          textTheme: ButtonTextTheme.primary,
          shape: StadiumBorder(),
          height: kMinInteractiveDimension,
        ),
      ),
      home: LoginScreen(),
    );
  }
}

class LoginScreen extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return LayoutBuilder(
      builder: (BuildContext context, BoxConstraints constraints) {
        return Scaffold(
          body: LoginForm(
            height: constraints.maxHeight,
          ),
        );
      },
    );
  }
}

class LoginForm extends StatelessWidget {
  const LoginForm({
    Key key,
    @required this.height,
  }) : super(key: key);

  final double height;

  @override
  Widget build(BuildContext context) {
    return Column(
      mainAxisAlignment: MainAxisAlignment.end,
      crossAxisAlignment: CrossAxisAlignment.stretch,
      children: <Widget>[
        CustomPaint(
          painter: _CurvePainter(
            height: height,
            foreground: Theme.of(context).primaryColor,
          ),
        ),
        Padding(
          padding: const EdgeInsets.symmetric(horizontal: 32.0),
          child: Column(
            crossAxisAlignment: CrossAxisAlignment.stretch,
            children: <Widget>[
              TextField(
                decoration: InputDecoration(
                  prefixIcon: Icon(Icons.person),
                ),
              ),
              SizedBox(height: 16.0),
              TextField(
                decoration: InputDecoration(
                  prefixIcon: Icon(Icons.lock),
                ),
              ),
              SizedBox(height: 16.0),
              RaisedButton(
                onPressed: () {},
                child: Text('SIGN IN'),
              ),
              SizedBox(height: 8.0),
              FlatButton(
                onPressed: () {},
                child: Text(
                  'Forgot Password?',
                  style: TextStyle(
                    color: Color(0xFF8fA4f7),
                    decoration: TextDecoration.underline,
                  ),
                  textAlign: TextAlign.left,
                ),
              ),
              SizedBox(height: 8.0),
              Row(
                children: <Widget>[
                  Expanded(
                    child: RaisedButton(
                      onPressed: () {},
                      shape: RoundedRectangleBorder(
                        borderRadius: BorderRadius.circular(2.0),
                      ),
                      color: Color(0xFF404988),
                      child: Text(
                        'FACEBOOK',
                        style: TextStyle(
                          fontSize: 18.0,
                          letterSpacing: 1.25,
                        ),
                      ),
                    ),
                  ),
                  const SizedBox(width: 4.0),
                  Expanded(
                    child: RaisedButton(
                      onPressed: () {},
                      shape: RoundedRectangleBorder(
                        borderRadius: BorderRadius.circular(2.0),
                      ),
                      color: Color(0xFF5C6EE8),
                      child: Text(
                        'GOOGLE',
                        style: TextStyle(
                          fontSize: 18.0,
                          letterSpacing: 1.25,
                        ),
                      ),
                    ),
                  ),
                ],
              ),
              SizedBox(height: 8.0),
              FlatButton(
                onPressed: () {},
                child: Text.rich(
                  TextSpan(
                    children: <TextSpan>[
                      TextSpan(
                        text: 'Don’t have an account? ',
                      ),
                      TextSpan(
                        text: 'Sign up',
                        style: TextStyle(
                          color: Color(0xFF8fA4f7),
                          decoration: TextDecoration.underline,
                        ),
                      ),
                    ],
                  ),
                  style: TextStyle(
                    color: Color(0xFF656565),
                  ),
                  textAlign: TextAlign.left,
                ),
              ),
              SizedBox(height: 32.0),
            ],
          ),
        ),
      ],
    );
  }
}

class _CurvePainter extends CustomPainter {
  const _CurvePainter({
    @required this.height,
    @required this.foreground,
  });

  final double height;
  final Color foreground;

  @override
  void paint(Canvas canvas, Size size) {
    final height20 = height / 5.0;
    final top = height20;
    final mid = height20 * 2.0;
    final bottom = height20 * 2.5;
    final center = size.width / 2.0;
    canvas.save();
    canvas.translate(0.0, -(height * 0.5) + 32.0);
    canvas.drawPath(
        Path()
          ..moveTo(0.0, top)
          ..quadraticBezierTo(0.0, mid, center, mid)
          ..quadraticBezierTo(size.width, mid, size.width, bottom)
          ..lineTo(size.width, -height)
          ..lineTo(0.0, -height)
          ..close(),
        Paint()
          ..style = PaintingStyle.fill
          ..color = foreground);
    canvas.restore();
  }

  @override
  bool shouldRepaint(_CurvePainter oldDelegate) {
    return height != oldDelegate.height || foreground != oldDelegate.foreground;
  }
}

@slightfoot woooooooooow, thanks a lot